{"id":2299,"date":"2019-05-09T15:54:20","date_gmt":"2019-05-09T15:54:20","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/?post_type=chapter&p=2299"},"modified":"2020-01-15T15:40:09","modified_gmt":"2020-01-15T15:40:09","slug":"1-5-section-exercises","status":"web-only","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/chapter\/1-5-section-exercises\/","title":{"raw":"1.5 Section Exercises","rendered":"1.5 Section Exercises"},"content":{"raw":"
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1.5 Section Exercises<\/h3>\r\n
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Verbal<\/h4>\r\n
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1. How does one find the domain of the quotient of two functions,[latex]\\text{ }\\frac{f}{g}?\\text{ }[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137836425\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137836425\"]\r\n

Find the numbers that make the function in the denominator[latex]\\text{ }g\\text{ }[\/latex]equal to zero, and check for any other domain restrictions on[latex]\\text{ }f\\text{ }[\/latex]and[latex]\\text{ }g,\\text{ }[\/latex]such as an even-indexed root or zeros in the denominator.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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2. What is the composition of two functions,[latex]\\text{ }f\\circ g?[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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\r\n\r\n3. If the order is reversed when composing two functions, can the result ever be the same as the answer in the original order of the composition? If yes, give an example. If no, explain why not.\r\n\r\n<\/div>\r\n
[reveal-answer q=\"fs-id1165135511306\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135511306\"]\r\n

Yes. Sample answer: Let[latex]\\text{ }f\\left(x\\right)=x+1\\text{ and }g\\left(x\\right)=x-1.\\text{ }[\/latex]Then[latex]\\text{ }f\\left(g\\left(x\\right)\\right)=f\\left(x-1\\right)=\\left(x-1\\right)+1=x\\text{ }[\/latex]and[latex]\\text{ }g\\left(f\\left(x\\right)\\right)=g\\left(x+1\\right)=\\left(x+1\\right)-1=x.\\text{ }[\/latex]So[latex]\\text{ }f\\circ g=g\\circ f.[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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4. How do you find the domain for the composition of two functions,[latex]\\text{ }f\\circ g?[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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Algebraic<\/h4>\r\n
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5. Given[latex]\\text{ }f\\left(x\\right)={x}^{2}+2x\\text{ }[\/latex]and[latex] g\\left(x\\right)=6-{x}^{2},[\/latex]find[latex]\\text{ }f+g,\\text{ }f-g,\\text{ }fg,\\text{ }[\/latex]and[latex]\\text{ }\\text{ }\\frac{f}{g}.[\/latex]Determine the domain for each function in interval notation.<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137645597\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137645597\"]\r\n

[latex]\\left(f+g\\right)\\left(x\\right)=2x+6,\\text{ }[\/latex]domain:[latex]\\text{ }\\left(-\\infty ,\\infty \\right)[\/latex]<\/p>\r\n

[latex]\\left(f-g\\right)\\left(x\\right)=2{x}^{2}+2x-6,\\text{ }[\/latex]domain:[latex]\\text{ }\\left(-\\infty ,\\infty \\right)[\/latex]<\/p>\r\n

[latex]\\left(fg\\right)\\left(x\\right)=-{x}^{4}-2{x}^{3}+6{x}^{2}+12x,\\text{ }[\/latex]domain:[latex]\\text{ }\\left(-\\infty ,\\infty \\right)[\/latex]<\/p>\r\n

[latex]\\left(\\frac{f}{g}\\right)\\left(x\\right)=\\frac{{x}^{2}+2x}{6-{x}^{2}},\\text{ }[\/latex]domain:[latex]\\text{ }\\left(-\\infty ,-\\sqrt{6}\\right)\\cup \\left(-\\sqrt{6},\\sqrt{6}\\right)\\cup \\left(\\sqrt{6},\\infty \\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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6. Given[latex]\\text{ }f\\left(x\\right)=-3{x}^{2}+x\\text{ }[\/latex]and[latex]\\text{ }g\\left(x\\right)=5,\\text{ }[\/latex]find[latex]\\text{ }f+g,\\text{ }f-g,\\text{ }fg,\\text{ }[\/latex]and[latex]\\text{ }\\text{ }\\frac{f}{g}.\\text{ }[\/latex]Determine the domain for each function in interval notation.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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7. Given[latex]\\text{ }f\\left(x\\right)=2{x}^{2}+4x\\text{ }[\/latex]and[latex]\\text{ }g\\left(x\\right)=\\frac{1}{2x},[\/latex]find[latex]\\text{ }f+g,\\text{ }f-g,\\text{ }fg,\\text{ }[\/latex]and[latex]\\text{ }\\frac{f}{g}.\\text{ }[\/latex]Determine the domain for each function in interval notation.<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137533872\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137533872\"]\r\n

[latex]\\left(f+g\\right)\\left(x\\right)=\\frac{4{x}^{3}+8{x}^{2}+1}{2x},\\text{ }[\/latex]domain:[latex]\\text{ }\\left(-\\infty ,0\\right)\\cup \\left(0,\\infty \\right)[\/latex]<\/p>\r\n

[latex]\\left(f-g\\right)\\left(x\\right)=\\frac{4{x}^{3}+8{x}^{2}-1}{2x},\\text{ }[\/latex]domain:[latex]\\text{ }\\left(-\\infty ,0\\right)\\cup \\left(0,\\infty \\right)[\/latex]<\/p>\r\n

[latex]\\left(fg\\right)\\left(x\\right)=x+2,\\text{ }[\/latex]domain:[latex]\\text{ }\\left(-\\infty ,0\\right)\\cup \\left(0,\\infty \\right)[\/latex]<\/p>\r\n

[latex]\\left(\\frac{f}{g}\\right)\\left(x\\right)=4{x}^{3}+8{x}^{2},\\text{ }[\/latex]domain:[latex]\\text{ }\\left(-\\infty ,0\\right)\\cup \\left(0,\\infty \\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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8. Given[latex]\\text{ }f\\left(x\\right)=\\frac{1}{x-4}\\text{ }[\/latex]and[latex]\\text{ }g\\left(x\\right)=\\frac{1}{6-x},\\text{ }[\/latex]find[latex]\\text{ }f+g,\\text{ }f-g,\\text{ }fg,\\text{ }[\/latex]and[latex]\\text{ }\\frac{f}{g}.\\text{ }[\/latex]Determine the domain for each function in interval notation.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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9. Given [latex]\\text{ }f\\left(x\\right)=3{x}^{2}\\text{ }[\/latex]and[latex]\\text{ }g\\left(x\\right)=\\sqrt{x-5},\\text{ }[\/latex]find[latex]\\text{ }f+g,\\text{ }f-g,\\text{ }fg,\\text{ }[\/latex]and[latex]\\text{ }\\frac{f}{g}.\\text{ }[\/latex]Determine the domain for each function in interval notation.<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137696948\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137696948\"]\r\n

[latex]\\left(f+g\\right)\\left(x\\right)=3{x}^{2}+\\sqrt{x-5},\\text{ }[\/latex]domain:[latex]\\text{ }\\left[5,\\infty \\right)[\/latex]<\/p>\r\n

[latex]\\left(f-g\\right)\\left(x\\right)=3{x}^{2}-\\sqrt{x-5},\\text{ }[\/latex]domain:[latex]\\text{ }\\left[5,\\infty \\right)[\/latex]<\/p>\r\n

[latex]\\left(fg\\right)\\left(x\\right)=3{x}^{2}\\sqrt{x-5},\\text{ }[\/latex]domain:[latex]\\text{ }\\left[5,\\infty \\right)[\/latex]<\/p>\r\n

[latex]\\left(\\frac{f}{g}\\right)\\left(x\\right)=\\frac{3{x}^{2}}{\\sqrt{x-5}},\\text{ }[\/latex]domain:[latex]\\text{ }\\left(5,\\infty \\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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10. Given[latex]\\text{ }f\\left(x\\right)=\\sqrt{x}\\text{ }[\/latex]and[latex]\\text{ }g\\left(x\\right)=|x-3|,\\text{ }[\/latex]find[latex]\\text{ }\\frac{g}{f}.\\text{ }[\/latex]Determine the domain of the function in interval notation.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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11. Given[latex]\\text{ }f\\left(x\\right)=2{x}^{2}+1\\text{ }[\/latex]and[latex]\\text{ }g\\left(x\\right)=3x-5,\\text{ }[\/latex]find the following:<\/p>\r\n\r\n

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  1. [latex]f\\left(g\\left(2\\right)\\right)[\/latex]<\/li>\r\n \t
  2. [latex]f\\left(g\\left(x\\right)\\right)[\/latex]<\/li>\r\n \t
  3. [latex]g\\left(f\\left(x\\right)\\right)[\/latex]<\/li>\r\n \t
  4. [latex]\\left(g\\circ g\\right)\\left(x\\right)[\/latex]<\/li>\r\n \t
  5. [latex]\\left(f\\circ f\\right)\\left(-2\\right)[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n
    [reveal-answer q=\"fs-id1165137526473\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137526473\"]\r\n

    a. 3; b.[latex]\\text{ }f\\left(g\\left(x\\right)\\right)=2{\\left(3x-5\\right)}^{2}+1;\\text{ }[\/latex]c.[latex]\\text{ }f\\left(g\\left(x\\right)\\right)=6{x}^{2}-2;\\text{ }[\/latex]d.[latex]\\text{ }\\left(g\\circ g\\right)\\left(x\\right)=3\\left(3x-5\\right)-5=9x-20;\\text{ }[\/latex]e.[latex]\\text{ }\\left(f\\circ f\\right)\\left(-2\\right)=163[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

    For the following exercises, use each pair of functions to find[latex]\\text{ }f\\left(g\\left(x\\right)\\right)\\text{ }[\/latex]and[latex]\\text{ }g\\left(f\\left(x\\right)\\right).\\text{ }[\/latex]Simplify your answers.<\/p>\r\n\r\n

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    12. [latex]f\\left(x\\right)={x}^{2}+1,\\text{ }g\\left(x\\right)=\\sqrt{x+2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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    13. [latex]f\\left(x\\right)=\\sqrt{x}+2,\\text{ }g\\left(x\\right)={x}^{2}+3[\/latex]<\/p>\r\n\r\n<\/div>\r\n

    [reveal-answer q=\"fs-id1165137407208\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137407208\"]\r\n

    [latex]f\\left(g\\left(x\\right)\\right)=\\sqrt{{x}^{2}+3}+2,\\text{ }g\\left(f\\left(x\\right)\\right)=x+4\\sqrt{x}+7[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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    14. [latex]f\\left(x\\right)=|x|,\\text{ }g\\left(x\\right)=5x+1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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    15. [latex]f\\left(x\\right)=\\sqrt[3]{x},\\text{ }g\\left(x\\right)=\\frac{x+1}{{x}^{3}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

    [reveal-answer q=\"fs-id1165137648072\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137648072\"]\r\n

    [latex]f\\left(g\\left(x\\right)\\right)=\\sqrt[3]{\\frac{x+1}{{x}^{3}}}=\\frac{\\sqrt[3]{x+1}}{x},\\text{ }g\\left(f\\left(x\\right)\\right)=\\frac{\\sqrt[3]{x}+1}{x}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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    16. [latex]f\\left(x\\right)=\\frac{1}{x-6},\\text{ }g\\left(x\\right)=\\frac{7}{x}+6[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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    17. [latex]f\\left(x\\right)=\\frac{1}{x-4},\\text{ }g\\left(x\\right)=\\frac{2}{x}+4[\/latex]<\/p>\r\n\r\n<\/div>\r\n

    [reveal-answer q=\"fs-id1165137638769\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137638769\"]\r\n

    [latex]\\left(f\\circ g\\right)\\left(x\\right)=\\frac{1}{\\frac{2}{x}+4-4}=\\frac{x}{2},\\text{ }\\left(g\\circ f\\right)\\left(x\\right)=2x-4[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

    For the following exercises, use each set of functions to find[latex]\\text{ }f\\left(g\\left(h\\left(x\\right)\\right)\\right).\\text{ }[\/latex]Simplify your answers.<\/p>\r\n\r\n

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    18. [latex]f\\left(x\\right)={x}^{4}+6,\\text{ }[\/latex][latex]g\\left(x\\right)=x-6,\\text{ }[\/latex]and[latex]\\text{ }h\\left(x\\right)=\\sqrt{x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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    19. [latex]f\\left(x\\right)={x}^{2}+1,\\text{ }[\/latex][latex]g\\left(x\\right)=\\frac{1}{x},\\text{ }[\/latex]and[latex]\\text{ }h\\left(x\\right)=x+3[\/latex]<\/p>\r\n\r\n<\/div>\r\n

    [reveal-answer q=\"fs-id1165137644526\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137644526\"]\r\n

    [latex]f\\left(g\\left(h\\left(x\\right)\\right)\\right)={\\left(\\frac{1}{x+3}\\right)}^{2}+1[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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    20. Given[latex]\\text{ }f\\left(x\\right)=\\frac{1}{x}\\text{ }[\/latex]and[latex]\\text{ }g\\left(x\\right)=x-3,\\text{ }[\/latex]find the following:<\/p>\r\n\r\n

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    1. [latex]\\left(f\\circ g\\right)\\left(x\\right)[\/latex]<\/li>\r\n \t
    2. the domain of[latex]\\text{ }\\left(f\\circ g\\right)\\left(x\\right)\\text{ }[\/latex]in interval notation<\/li>\r\n \t
    3. [latex]\\left(g\\circ f\\right)\\left(x\\right)[\/latex]<\/li>\r\n \t
    4. the domain of[latex]\\text{ }\\left(g\\circ f\\right)\\left(x\\right)\\text{ }[\/latex]<\/li>\r\n \t
    5. [latex]\\left(\\frac{f}{g}\\right)x[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n
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      21. Given[latex]\\text{ }f\\left(x\\right)=\\sqrt{2-4x}\\text{ }[\/latex]and[latex]\\text{ }g\\left(x\\right)=-\\frac{3}{x},\\text{ }[\/latex]find the following:<\/p>\r\n\r\n

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      1. [latex]\\left(g\\circ f\\right)\\left(x\\right)[\/latex]<\/li>\r\n \t
      2. the domain of[latex]\\text{ }\\left(g\\circ f\\right)\\left(x\\right)\\text{ }[\/latex]in interval notation<\/li>\r\n<\/ol>\r\n<\/div>\r\n
        [reveal-answer q=\"fs-id1165137722654\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137722654\"]\r\n

        a.[latex]\\text{ }\\left(g\\circ f\\right)\\left(x\\right)=-\\frac{3}{\\sqrt{2-4x}};\\text{ }[\/latex]b.[latex]\\text{ }\\left(-\\infty ,\\frac{1}{2}\\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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        22. Given the functions[latex]\\text{ }f\\left(x\\right)=\\frac{1-x}{x}\\text{ }\\text{and}\\text{ }g\\left(x\\right)=\\frac{1}{1+{x}^{2}},\\text{ }[\/latex] find the following:<\/p>\r\n\r\n

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        1. [latex]\\left(g\\circ f\\right)\\left(x\\right)[\/latex]<\/li>\r\n \t
        2. [latex]\\left(g\\circ f\\right)\\left(\\text{2}\\right)[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n
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          23. Given functions[latex]\\text{ }p\\left(x\\right)=\\frac{1}{\\sqrt{x}}\\text{ }[\/latex]and[latex]\\text{ }m\\left(x\\right)={x}^{2}-4,\\text{ }[\/latex]state the domain of each of the following functions using interval notation:<\/p>\r\n\r\n

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          1. [latex]\\frac{p\\left(x\\right)}{m\\left(x\\right)}[\/latex]<\/li>\r\n \t
          2. [latex]p\\left(m\\left(x\\right)\\right)[\/latex]<\/li>\r\n \t
          3. [latex]m\\left(p\\left(x\\right)\\right)[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n
            [reveal-answer q=\"fs-id1165137387082\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137387082\"]\r\n

            a.[latex]\\text{ }\\left(0,2\\right)\\cup \\left(2,\\infty \\right);\\text{ }[\/latex]b.[latex]\\text{ }\\left(-\\infty ,-2\\right)\\cup \\left(2,\\infty \\right);\\text{ }[\/latex]c.[latex]\\text{ }\\left(0,\\infty \\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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            24. Given functions[latex]\\text{ }q\\left(x\\right)=\\frac{1}{\\sqrt{x}}\\text{ }[\/latex]and[latex]\\text{ }h\\left(x\\right)={x}^{2}-9,\\text{ }[\/latex]state the domain of each of the following functions using interval notation.<\/p>\r\n\r\n

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            1. [latex]\\frac{q\\left(x\\right)}{h\\left(x\\right)}[\/latex]<\/li>\r\n \t
            2. [latex]q\\left(h\\left(x\\right)\\right)[\/latex]<\/li>\r\n \t
            3. [latex]h\\left(q\\left(x\\right)\\right)[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n
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              25. For[latex]\\text{ }f\\left(x\\right)=\\frac{1}{x}\\text{ }[\/latex]and[latex]\\text{ }g\\left(x\\right)=\\sqrt{x-1},\\text{ }[\/latex]write the domain of[latex]\\text{ }\\left(f\\circ g\\right)\\left(x\\right)\\text{ }[\/latex]in interval notation.<\/p>\r\n\r\n<\/div>\r\n

              [reveal-answer q=\"fs-id1165137889762\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137889762\"]\r\n

              [latex]\\left(1,\\infty \\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

              For the following exercises, find functions[latex]\\text{ }f\\left(x\\right)\\text{ }[\/latex]and[latex]\\text{ }g\\left(x\\right)\\text{ }[\/latex]so the given function can be expressed as[latex]\\text{ }h\\left(x\\right)=f\\left(g\\left(x\\right)\\right).[\/latex]<\/p>\r\n\r\n

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              26. [latex]h\\left(x\\right)={\\left(x+2\\right)}^{2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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              27. [latex]h\\left(x\\right)={\\left(x-5\\right)}^{3}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

              [reveal-answer q=\"fs-id1165137432927\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137432927\"]\r\n

              sample:[latex]\\begin{array}{l}f\\left(x\\right)={x}^{3}\\\\ g\\left(x\\right)=x-5\\end{array}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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              28. [latex]h\\left(x\\right)=\\frac{3}{x-5}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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              29. [latex]h\\left(x\\right)=\\frac{4}{{\\left(x+2\\right)}^{2}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

              [reveal-answer q=\"fs-id1165137598944\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137598944\"]\r\n

              sample:[latex]\\begin{array}{l}f\\left(x\\right)=\\frac{4}{x}\\hfill \\\\ g\\left(x\\right)={\\left(x+2\\right)}^{2}\\hfill \\end{array}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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              30. [latex]h\\left(x\\right)=4+\\sqrt[3]{x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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              31. [latex]h\\left(x\\right)=\\sqrt[3]{\\frac{1}{2x-3}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

              [reveal-answer q=\"fs-id1165135699087\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135699087\"]\r\n

              sample:[latex]\\begin{array}{l}f\\left(x\\right)=\\sqrt[3]{x}\\\\ g\\left(x\\right)=\\frac{1}{2x-3}\\end{array}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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              32. [latex]h\\left(x\\right)=\\frac{1}{{\\left(3{x}^{2}-4\\right)}^{-3}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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              \r\n

              33.[latex]h\\left(x\\right)=\\sqrt[4]{\\frac{3x-2}{x+5}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

              [reveal-answer q=\"fs-id1165137731127\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137731127\"]\r\n

              sample:[latex]\\begin{array}{l}f\\left(x\\right)=\\sqrt[4]{x}\\\\ g\\left(x\\right)=\\frac{3x-2}{x+5}\\end{array}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

              \r\n
              \r\n

              34. [latex]h\\left(x\\right)={\\left(\\frac{8+{x}^{3}}{8-{x}^{3}}\\right)}^{4}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

              \r\n
              \r\n

              35. [latex]h\\left(x\\right)=\\sqrt{2x+6}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

              [reveal-answer q=\"fs-id1165135439855\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135439855\"]\r\n

              sample: [latex]f\\left(x\\right)=\\sqrt{x}[\/latex]<\/p>\r\n\r\n

              [\/hidden-answer]<\/div>\r\n
              <\/div>\r\n
              36. [latex]g\\left(x\\right)=2x+6[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
              \r\n
              \r\n

              37. [latex]h\\left(x\\right)={\\left(5x-1\\right)}^{3}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

              \r\n
              \r\n

              38. [latex]h\\left(x\\right)=\\sqrt[3]{x-1}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

              [reveal-answer q=\"fs-id1165137806156\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137806156\"]\r\n

              sample: [latex]f\\left(x\\right)=\\sqrt[3]{x}[\/latex]<\/p>\r\n\r\n

              [\/hidden-answer]<\/div>\r\n39. [latex]g\\left(x\\right)=\\left(x-1\\right)[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n
              \r\n
              \r\n

              40. [latex]h\\left(x\\right)=|{x}^{2}+7|[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

              \r\n
              \r\n

              41. [latex]h\\left(x\\right)=\\frac{1}{{\\left(x-2\\right)}^{3}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

              [reveal-answer q=\"fs-id1165135708080\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135708080\"]\r\n

              sample: [latex]f\\left(x\\right)={x}^{3}[\/latex]<\/p>\r\n\r\n

              [\/hidden-answer]<\/div>\r\n42. [latex]g\\left(x\\right)=\\frac{1}{x-2}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n
              \r\n
              \r\n

              43. [latex]h\\left(x\\right)={\\left(\\frac{1}{2x-3}\\right)}^{2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

              \r\n
              \r\n

              44. [latex]h\\left(x\\right)=\\sqrt{\\frac{2x-1}{3x+4}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

              [reveal-answer q=\"fs-id1165137596478\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137596478\"]\r\n

              sample: [latex]f\\left(x\\right)=\\sqrt{x}[\/latex]<\/p>\r\n\r\n

              [\/hidden-answer]<\/div>\r\n45. [latex]g\\left(x\\right)=\\frac{2x-1}{3x+4}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n
              \r\n

              Graphical<\/h4>\r\n

              For the following exercises, use the graphs of[latex]\\text{ }f,[\/latex]shown in (Figure)<\/a>, and[latex]\\text{ }g,[\/latex]shown in (Figure)<\/a>, to evaluate the expressions.<\/p>\r\n\r\n

              \"graph<\/span><\/div>\r\n
              \u00a0<\/span><\/div>\r\n
              \"\"<\/div>\r\n
              \r\n
              \r\n

              46. [latex]f\\left(g\\left(3\\right)\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

              \r\n
              \r\n

              47. [latex]f\\left(g\\left(1\\right)\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

              [reveal-answer q=\"fs-id1165135650744\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135650744\"]\r\n

              2<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

              \r\n
              \r\n

              48. [latex]g\\left(f\\left(1\\right)\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

              \r\n
              \r\n

              49. [latex]g\\left(f\\left(0\\right)\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

              [reveal-answer q=\"fs-id1165137759779\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137759779\"]\r\n

              5<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

              \r\n
              \r\n

              50. [latex]f\\left(f\\left(5\\right)\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

              \r\n
              \r\n

              51. [latex]f\\left(f\\left(4\\right)\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

              [reveal-answer q=\"fs-id1165134394616\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134394616\"]\r\n

              4<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

              \r\n
              \r\n

              52. [latex]g\\left(g\\left(2\\right)\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

              \r\n
              \r\n

              53. [latex]g\\left(g\\left(0\\right)\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

              [reveal-answer q=\"fs-id1165131969120\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165131969120\"]\r\n

              0<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

              For the following exercises, use graphs of[latex]\\text{ }f\\left(x\\right),[\/latex]shown in (Figure)<\/a>,[latex]\\text{ }g\\left(x\\right),[\/latex]shown in (Figure)<\/a>, and[latex]\\text{ }h\\left(x\\right),[\/latex]shown in (Figure)<\/a>, to evaluate the expressions.<\/p>\r\n\"graph\r\n\r\n\"\"\r\n\r\n\"\"\r\n

              <\/div>\r\n
              <\/div>\r\n
              \r\n
              \r\n

              54. [latex]g\\left(f\\left(1\\right)\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

              \r\n
              \r\n

              55. [latex]g\\left(f\\left(2\\right)\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

              [reveal-answer q=\"fs-id1165135641698\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135641698\"]\r\n

              2<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

              \r\n
              \r\n

              56. [latex]f\\left(g\\left(4\\right)\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

              \r\n
              \r\n

              57. [latex]f\\left(g\\left(1\\right)\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

              [reveal-answer q=\"fs-id1165137694045\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137694045\"]\r\n

              1<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

              \r\n
              \r\n

              58. [latex]f\\left(h\\left(2\\right)\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

              \r\n
              \r\n

              59. [latex]h\\left(f\\left(2\\right)\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

              [reveal-answer q=\"fs-id1165137666936\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137666936\"]\r\n

              4<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

              \r\n
              \r\n

              60. [latex]f\\left(g\\left(h\\left(4\\right)\\right)\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

              \r\n
              \r\n

              61. [latex]f\\left(g\\left(f\\left(-2\\right)\\right)\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

              [reveal-answer q=\"fs-id1165137501367\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137501367\"]\r\n

              4<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

              \r\n

              Numeric<\/h4>\r\n

              For the following exercises, use the function values for[latex]\\text{ }f\\text{ and }g\\text{ }[\/latex]shown in (Figure)<\/a> to evaluate each expression.<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
              [latex]x[\/latex]<\/strong><\/td>\r\n[latex]f\\left(x\\right)[\/latex]<\/strong><\/td>\r\n[latex]g\\left(x\\right)[\/latex]<\/strong><\/td>\r\n<\/tr>\r\n
              0<\/td>\r\n7<\/td>\r\n9<\/td>\r\n<\/tr>\r\n
              1<\/td>\r\n6<\/td>\r\n5<\/td>\r\n<\/tr>\r\n
              2<\/td>\r\n5<\/td>\r\n6<\/td>\r\n<\/tr>\r\n
              3<\/td>\r\n8<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
              4<\/td>\r\n4<\/td>\r\n1<\/td>\r\n<\/tr>\r\n
              5<\/td>\r\n0<\/td>\r\n8<\/td>\r\n<\/tr>\r\n
              6<\/td>\r\n2<\/td>\r\n7<\/td>\r\n<\/tr>\r\n
              7<\/td>\r\n1<\/td>\r\n3<\/td>\r\n<\/tr>\r\n
              8<\/td>\r\n9<\/td>\r\n4<\/td>\r\n<\/tr>\r\n
              9<\/td>\r\n3<\/td>\r\n0<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n
              \r\n
              \r\n

              62. [latex]f\\left(g\\left(8\\right)\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

              \r\n
              \r\n\r\n63. [latex]f\\left(g\\left(5\\right)\\right)[\/latex]\r\n\r\n<\/div>\r\n
              [reveal-answer q=\"fs-id1165134061990\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134061990\"]\r\n

              9<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

              \r\n
              \r\n

              64. [latex]g\\left(f\\left(5\\right)\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

              \r\n
              \r\n

              65. [latex]g\\left(f\\left(3\\right)\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

              [reveal-answer q=\"fs-id1165133023676\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165133023676\"]\r\n

              4<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

              \r\n
              \r\n

              66. [latex]f\\left(f\\left(4\\right)\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

              \r\n
              \r\n

              67. [latex]f\\left(f\\left(1\\right)\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

              [reveal-answer q=\"fs-id1165135417784\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135417784\"]\r\n

              2<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

              \r\n
              \r\n

              68. [latex]g\\left(g\\left(2\\right)\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

              \r\n
              \r\n

              69. [latex]g\\left(g\\left(6\\right)\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

              [reveal-answer q=\"fs-id1165135457908\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135457908\"]\r\n

              3<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

              For the following exercises, use the function values for[latex]\\text{ }f\\text{ and }g\\text{ }[\/latex]shown in (Figure)<\/a> to evaluate the expressions.<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
              [latex]x[\/latex]<\/strong><\/td>\r\n[latex]f\\left(x\\right)[\/latex]<\/strong><\/td>\r\n[latex]g\\left(x\\right)[\/latex]<\/strong><\/td>\r\n<\/tr>\r\n
              -3<\/td>\r\n11<\/td>\r\n-8<\/td>\r\n<\/tr>\r\n
              -2<\/td>\r\n9<\/td>\r\n-3<\/td>\r\n<\/tr>\r\n
              -1<\/td>\r\n7<\/td>\r\n0<\/td>\r\n<\/tr>\r\n
              0<\/td>\r\n5<\/td>\r\n1<\/td>\r\n<\/tr>\r\n
              1<\/td>\r\n3<\/td>\r\n0<\/td>\r\n<\/tr>\r\n
              2<\/td>\r\n1<\/td>\r\n-3<\/td>\r\n<\/tr>\r\n
              3<\/td>\r\n-1<\/td>\r\n-8<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n
              \r\n
              \r\n

              70. [latex]\\left(f\\circ g\\right)\\left(1\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

              \r\n
              \r\n

              71. [latex]\\left(f\\circ g\\right)\\left(2\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

              [reveal-answer q=\"fs-id1165133047570\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165133047570\"]\r\n

              11<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

              \r\n
              \r\n

              72. [latex]\\left(g\\circ f\\right)\\left(2\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

              \r\n
              \r\n

              73. [latex]\\left(g\\circ f\\right)\\left(3\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

              [reveal-answer q=\"fs-id1165137427576\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137427576\"]\r\n

              0<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

              \r\n
              \r\n

              74. [latex]\\left(g\\circ g\\right)\\left(1\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

              \r\n
              \r\n

              75. [latex]\\left(f\\circ f\\right)\\left(3\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

              [reveal-answer q=\"fs-id1165137626894\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137626894\"]\r\n

              7<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

              For the following exercises, use each pair of functions to find[latex]\\text{ }f\\left(g\\left(0\\right)\\right)\\text{ }[\/latex]and[latex]\\text{ }g\\left(f\\left(0\\right)\\right).[\/latex]<\/p>\r\n\r\n

              \r\n
              \r\n

              76. [latex]f\\left(x\\right)=4x+8,\\text{ }g\\left(x\\right)=7-{x}^{2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

              \r\n
              \r\n

              77. [latex]f\\left(x\\right)=5x+7,\\text{ }g\\left(x\\right)=4-2{x}^{2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

              [reveal-answer q=\"fs-id1165135657107\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135657107\"]\r\n

              [latex]f\\left(g\\left(0\\right)\\right)=27,\\text{ }g\\left(f\\left(0\\right)\\right)=-94[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

              \r\n
              \r\n

              78. [latex]f\\left(x\\right)=\\sqrt{x+4},\\text{ }g\\left(x\\right)=12-{x}^{3}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

              \r\n
              \r\n

              79. [latex]f\\left(x\\right)=\\frac{1}{x+2},\\text{ }g\\left(x\\right)=4x+3[\/latex]<\/p>\r\n\r\n<\/div>\r\n

              [reveal-answer q=\"fs-id1165137767238\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137767238\"]\r\n

              [latex]f\\left(g\\left(0\\right)\\right)=\\frac{1}{5},\\text{ }g\\left(f\\left(0\\right)\\right)=5[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

              For the following exercises, use the functions[latex]\\text{ }f\\left(x\\right)=2{x}^{2}+1\\text{ }[\/latex]and[latex]\\text{ }g\\left(x\\right)=3x+5\\text{ }[\/latex] to evaluate or find the composite function as indicated.<\/p>\r\n\r\n

              \r\n
              \r\n

              80. [latex]f\\left(g\\left(2\\right)\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

              \r\n
              \r\n

              81. [latex]f\\left(g\\left(x\\right)\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

              [reveal-answer q=\"fs-id1165137480928\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137480928\"]\r\n

              [latex]18{x}^{2}+60x+51[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

              \r\n
              \r\n

              82. [latex]g\\left(f\\left(-3\\right)\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

              \r\n
              \r\n

              83. [latex]\\left(g\\circ g\\right)\\left(x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

              [reveal-answer q=\"fs-id1165137758394\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137758394\"]\r\n

              [latex]g\\circ g\\left(x\\right)=9x+20[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

              \r\n

              Extensions<\/h4>\r\n

              For the following exercises, use[latex]\\text{ }f\\left(x\\right)={x}^{3}+1\\text{ }[\/latex]and[latex]\\text{ }g\\left(x\\right)=\\sqrt[3]{x-1}.[\/latex]<\/p>\r\n\r\n

              \r\n
              \r\n

              84. Find[latex]\\text{ }\\left(f\\circ g\\right)\\left(x\\right)\\text{ }[\/latex]and[latex]\\text{ }\\left(g\\circ f\\right)\\left(x\\right).\\text{ }[\/latex]Compare the two answers.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

              \r\n
              \r\n

              85. Find[latex]\\text{ }\\left(f\\circ g\\right)\\left(2\\right)\\text{ }[\/latex]and[latex]\\text{ }\\left(g\\circ f\\right)\\left(2\\right).[\/latex]<\/p>\r\n\r\n<\/div>\r\n

              [reveal-answer q=\"fs-id1165135198523\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135198523\"]\r\n

              2<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

              \r\n
              \r\n

              86. What is the domain of[latex]\\text{ }\\left(g\\circ f\\right)\\left(x\\right)?[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

              \r\n
              \r\n

              87. What is the domain of[latex]\\text{ }\\left(f\\circ g\\right)\\left(x\\right)?[\/latex]<\/p>\r\n\r\n<\/div>\r\n

              [reveal-answer q=\"fs-id1165133190973\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165133190973\"]\r\n

              [latex]\\left(-\\infty ,\\infty \\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

              \r\n
              \r\n

              88. Let[latex]\\text{ }f\\left(x\\right)=\\frac{1}{x}.[\/latex]<\/p>\r\n\r\n

                \r\n \t
              1. Find[latex]\\text{ }\\left(f\\circ f\\right)\\left(x\\right).[\/latex]<\/li>\r\n \t
              2. Is[latex]\\text{ }\\left(f\\circ f\\right)\\left(x\\right)\\text{ }[\/latex]for any function[latex]\\text{ }f\\text{ }[\/latex]the same result as the answer to part (a) for any function? Explain.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n

                For the following exercises, let[latex]\\text{ }F\\left(x\\right)={\\left(x+1\\right)}^{5},\\text{ }[\/latex][latex]f\\left(x\\right)={x}^{5},\\text{ }[\/latex]and[latex]\\text{ }g\\left(x\\right)=x+1.[\/latex]<\/p>\r\n\r\n

                \r\n
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                89. True or False:[latex]\\text{ }\\left(g\\circ f\\right)\\left(x\\right)=F\\left(x\\right).[\/latex]<\/p>\r\n\r\n<\/div>\r\n

                [reveal-answer q=\"fs-id1165137837867\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137837867\"]False[\/hidden-answer]<\/div>\r\n<\/div>\r\n
                \r\n
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                90. True or False:[latex]\\text{ }\\left(f\\circ g\\right)\\left(x\\right)=F\\left(x\\right).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

                For the following exercises, find the composition when[latex]\\text{ }f\\left(x\\right)={x}^{2}+2\\text{ }[\/latex]for all[latex]\\text{ }x\\ge 0\\text{ }[\/latex]and[latex]\\text{ }g\\left(x\\right)=\\sqrt{x-2}.[\/latex]<\/p>\r\n\r\n

                \r\n
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                91. [latex]\\left(f\\circ g\\right)\\left(6\\right);\\text{ }\\left(g\\circ f\\right)\\left(6\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

                [reveal-answer q=\"fs-id1165137939500\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137939500\"]\r\n

                [latex]\\left(f\\circ g\\right)\\left(6\\right)=6[\/latex]; [latex]\\text{ }\\left(g\\circ f\\right)\\left(6\\right)=6[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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                92. [latex]\\left(g\\circ f\\right)\\left(a\\right);\\text{ }\\left(f\\circ g\\right)\\left(a\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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                93. [latex]\\left(f\\circ g\\right)\\left(11\\right);\\text{ }\\left(g\\circ f\\right)\\left(11\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

                [reveal-answer q=\"fs-id1165137823068\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137823068\"]\r\n

                [latex]\\left(f\\circ g\\right)\\left(11\\right)=11\\text{ },\\text{ }\\left(g\\circ f\\right)\\left(11\\right)=11[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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                Real-World Applications<\/h4>\r\n
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                94. The function[latex]\\text{ }D\\left(p\\right)\\text{ }[\/latex]gives the number of items that will be demanded when the price is[latex]\\text{ }p.\\text{ }[\/latex]The production cost[latex]\\text{ }C\\left(x\\right)\\text{ }[\/latex]is the cost of producing[latex]\\text{ }x\\text{ }[\/latex]items. To determine the cost of production when the price is $6, you would do which of the following?<\/p>\r\n\r\n

                  \r\n \t
                1. Evaluate[latex]\\text{ }D\\left(C\\left(6\\right)\\right).[\/latex]<\/li>\r\n \t
                2. Evaluate[latex]\\text{ }C\\left(D\\left(6\\right)\\right).[\/latex]<\/li>\r\n \t
                3. Solve[latex]\\text{ }D\\left(C\\left(x\\right)\\right)=6.[\/latex]<\/li>\r\n \t
                4. Solve[latex]\\text{ }C\\left(D\\left(p\\right)\\right)=6.[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n
                  \r\n
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                  95. The function[latex]\\text{ }A\\left(d\\right)\\text{ }[\/latex]gives the pain level on a scale of 0 to 10 experienced by a patient with[latex]\\text{ }d\\text{ }[\/latex]milligrams of a pain-reducing drug in her system. The milligrams of the drug in the patient\u2019s system after[latex]\\text{ }t\\text{ }[\/latex]minutes is modeled by[latex]\\text{ }m\\left(t\\right).\\text{ }[\/latex]Which of the following would you do in order to determine when the patient will be at a pain level of 4?<\/p>\r\n\r\n

                    \r\n \t
                  1. Evaluate[latex]\\text{ }A\\left(m\\left(4\\right)\\right).[\/latex]<\/li>\r\n \t
                  2. Evaluate[latex]\\text{ }m\\left(A\\left(4\\right)\\right).[\/latex]<\/li>\r\n \t
                  3. Solve[latex]\\text{ }A\\left(m\\left(t\\right)\\right)=4.[\/latex]<\/li>\r\n \t
                  4. Solve[latex]\\text{ }m\\left(A\\left(d\\right)\\right)=4.[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n
                    [reveal-answer q=\"fs-id1165137419674\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137419674\"]\r\n

                    c<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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                    96. A store offers customers a 30% discount on the price[latex]\\text{ }x\\text{ }[\/latex]of selected items. Then, the store takes off an additional 15% at the cash register. Write a price function[latex]\\text{ }P\\left(x\\right)\\text{ }[\/latex]that computes the final price of the item in terms of the original price[latex]\\text{ }x.\\text{ }[\/latex](Hint: Use function composition to find your answer.)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

                    \r\n
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                    97. A rain drop hitting a lake makes a circular ripple. If the radius, in inches, grows as a function of time in minutes according to[latex]\\text{ }r\\left(t\\right)=25\\sqrt{t+2},\\text{ }[\/latex]find the area of the ripple as a function of time. Find the area of the ripple at[latex]\\text{ }t=2.[\/latex]<\/p>\r\n\r\n<\/div>\r\n

                    [reveal-answer q=\"fs-id1165135530544\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135530544\"]\r\n

                    [latex]A\\left(t\\right)=\\pi {\\left(25\\sqrt{t+2}\\right)}^{2}\\text{ }[\/latex]and[latex]\\text{ }A\\left(2\\right)=\\pi {\\left(25\\sqrt{4}\\right)}^{2}=2500\\pi [\/latex] square inches<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

                    \r\n
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                    98. A forest fire leaves behind an area of grass burned in an expanding circular pattern. If the radius of the circle of burning grass is increasing with time according to the formula[latex]\\text{ }r\\left(t\\right)=2t+1,\\text{ }[\/latex]express the area burned as a function of time,[latex]\\text{ }t\\text{ }[\/latex](minutes).<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

                    \r\n
                    \r\n

                    99. Use the function you found in the previous exercise to find the total area burned after 5 minutes.<\/p>\r\n\r\n<\/div>\r\n

                    [reveal-answer q=\"fs-id1165134031328\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134031328\"]\r\n

                    [latex]A\\left(5\\right)=\\pi {\\left(2\\left(5\\right)+1\\right)}^{2}=121\\pi \\text{ }[\/latex]square units<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

                    \r\n
                    \r\n

                    100. The radius[latex]\\text{ }r,\\text{ }[\/latex]in inches, of a spherical balloon is related to the volume,[latex]\\text{ }V,\\text{ }[\/latex]by[latex]\\text{ }r\\left(V\\right)=\\sqrt[3]{\\frac{3V}{4\\pi }}.\\text{ }[\/latex]Air is pumped into the balloon, so the volume after[latex]\\text{ }t\\text{ }[\/latex]seconds is given by[latex]\\text{ }V\\left(t\\right)=10+20t.[\/latex]<\/p>\r\n\r\n

                      \r\n \t
                    1. Find the composite function[latex]\\text{ }r\\left(V\\left(t\\right)\\right).[\/latex]<\/li>\r\n \t
                    2. Find the exact<\/em> time when the radius reaches 10 inches.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n
                      \r\n
                      \r\n

                      101. The number of bacteria in a refrigerated food product is given by [latex]N\\left(T\\right)=23{T}^{2}-56T+1,\\text{ }[\/latex][latex]3<T<33,\\text{ }[\/latex]where[latex]\\text{ }T[\/latex] is the temperature of the food. When the food is removed from the refrigerator, the temperature is given by [latex]T\\left(t\\right)=5t+1.5,[\/latex] where [latex]t[\/latex] is the time in hours.<\/p>\r\n\r\n

                        \r\n \t
                      1. Find the composite function[latex]\\text{ }N\\left(T\\left(t\\right)\\right).[\/latex]<\/li>\r\n \t
                      2. Find the time (round to two decimal places) when the bacteria count reaches 6752.<\/li>\r\n<\/ol>\r\n<\/div>\r\n
                        [reveal-answer q=\"fs-id1165134107339\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134107339\"]\r\n

                        a.[latex]\\text{ }N\\left(T\\left(t\\right)\\right)=23{\\left(5t+1.5\\right)}^{2}-56\\left(5t+1.5\\right)+1;\\text{ }[\/latex] b. 3.38 hours<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n ","rendered":"

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                        1.5 Section Exercises<\/h3>\n
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                        Verbal<\/h4>\n
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                        1. How does one find the domain of the quotient of two functions,[latex]\\text{ }\\frac{f}{g}?\\text{ }[\/latex]<\/p>\n<\/div>\n

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                        Show Solution<\/span><\/p>\n
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                        Find the numbers that make the function in the denominator[latex]\\text{ }g\\text{ }[\/latex]equal to zero, and check for any other domain restrictions on[latex]\\text{ }f\\text{ }[\/latex]and[latex]\\text{ }g,\\text{ }[\/latex]such as an even-indexed root or zeros in the denominator.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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                        2. What is the composition of two functions,[latex]\\text{ }f\\circ g?[\/latex]<\/p>\n<\/div>\n<\/div>\n

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                        3. If the order is reversed when composing two functions, can the result ever be the same as the answer in the original order of the composition? If yes, give an example. If no, explain why not.<\/p>\n<\/div>\n

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                        Show Solution<\/span><\/p>\n
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                        Yes. Sample answer: Let[latex]\\text{ }f\\left(x\\right)=x+1\\text{ and }g\\left(x\\right)=x-1.\\text{ }[\/latex]Then[latex]\\text{ }f\\left(g\\left(x\\right)\\right)=f\\left(x-1\\right)=\\left(x-1\\right)+1=x\\text{ }[\/latex]and[latex]\\text{ }g\\left(f\\left(x\\right)\\right)=g\\left(x+1\\right)=\\left(x+1\\right)-1=x.\\text{ }[\/latex]So[latex]\\text{ }f\\circ g=g\\circ f.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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                        4. How do you find the domain for the composition of two functions,[latex]\\text{ }f\\circ g?[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n

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                        Algebraic<\/h4>\n
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                        5. Given[latex]\\text{ }f\\left(x\\right)={x}^{2}+2x\\text{ }[\/latex]and[latex]g\\left(x\\right)=6-{x}^{2},[\/latex]find[latex]\\text{ }f+g,\\text{ }f-g,\\text{ }fg,\\text{ }[\/latex]and[latex]\\text{ }\\text{ }\\frac{f}{g}.[\/latex]Determine the domain for each function in interval notation.<\/p>\n<\/div>\n

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                        Show Solution<\/span><\/p>\n
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                        [latex]\\left(f+g\\right)\\left(x\\right)=2x+6,\\text{ }[\/latex]domain:[latex]\\text{ }\\left(-\\infty ,\\infty \\right)[\/latex]<\/p>\n

                        [latex]\\left(f-g\\right)\\left(x\\right)=2{x}^{2}+2x-6,\\text{ }[\/latex]domain:[latex]\\text{ }\\left(-\\infty ,\\infty \\right)[\/latex]<\/p>\n

                        [latex]\\left(fg\\right)\\left(x\\right)=-{x}^{4}-2{x}^{3}+6{x}^{2}+12x,\\text{ }[\/latex]domain:[latex]\\text{ }\\left(-\\infty ,\\infty \\right)[\/latex]<\/p>\n

                        [latex]\\left(\\frac{f}{g}\\right)\\left(x\\right)=\\frac{{x}^{2}+2x}{6-{x}^{2}},\\text{ }[\/latex]domain:[latex]\\text{ }\\left(-\\infty ,-\\sqrt{6}\\right)\\cup \\left(-\\sqrt{6},\\sqrt{6}\\right)\\cup \\left(\\sqrt{6},\\infty \\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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                        6. Given[latex]\\text{ }f\\left(x\\right)=-3{x}^{2}+x\\text{ }[\/latex]and[latex]\\text{ }g\\left(x\\right)=5,\\text{ }[\/latex]find[latex]\\text{ }f+g,\\text{ }f-g,\\text{ }fg,\\text{ }[\/latex]and[latex]\\text{ }\\text{ }\\frac{f}{g}.\\text{ }[\/latex]Determine the domain for each function in interval notation.<\/p>\n<\/div>\n<\/div>\n

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                        7. Given[latex]\\text{ }f\\left(x\\right)=2{x}^{2}+4x\\text{ }[\/latex]and[latex]\\text{ }g\\left(x\\right)=\\frac{1}{2x},[\/latex]find[latex]\\text{ }f+g,\\text{ }f-g,\\text{ }fg,\\text{ }[\/latex]and[latex]\\text{ }\\frac{f}{g}.\\text{ }[\/latex]Determine the domain for each function in interval notation.<\/p>\n<\/div>\n

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                        Show Solution<\/span><\/p>\n
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                        [latex]\\left(f+g\\right)\\left(x\\right)=\\frac{4{x}^{3}+8{x}^{2}+1}{2x},\\text{ }[\/latex]domain:[latex]\\text{ }\\left(-\\infty ,0\\right)\\cup \\left(0,\\infty \\right)[\/latex]<\/p>\n

                        [latex]\\left(f-g\\right)\\left(x\\right)=\\frac{4{x}^{3}+8{x}^{2}-1}{2x},\\text{ }[\/latex]domain:[latex]\\text{ }\\left(-\\infty ,0\\right)\\cup \\left(0,\\infty \\right)[\/latex]<\/p>\n

                        [latex]\\left(fg\\right)\\left(x\\right)=x+2,\\text{ }[\/latex]domain:[latex]\\text{ }\\left(-\\infty ,0\\right)\\cup \\left(0,\\infty \\right)[\/latex]<\/p>\n

                        [latex]\\left(\\frac{f}{g}\\right)\\left(x\\right)=4{x}^{3}+8{x}^{2},\\text{ }[\/latex]domain:[latex]\\text{ }\\left(-\\infty ,0\\right)\\cup \\left(0,\\infty \\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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                        8. Given[latex]\\text{ }f\\left(x\\right)=\\frac{1}{x-4}\\text{ }[\/latex]and[latex]\\text{ }g\\left(x\\right)=\\frac{1}{6-x},\\text{ }[\/latex]find[latex]\\text{ }f+g,\\text{ }f-g,\\text{ }fg,\\text{ }[\/latex]and[latex]\\text{ }\\frac{f}{g}.\\text{ }[\/latex]Determine the domain for each function in interval notation.<\/p>\n<\/div>\n<\/div>\n

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                        9. Given [latex]\\text{ }f\\left(x\\right)=3{x}^{2}\\text{ }[\/latex]and[latex]\\text{ }g\\left(x\\right)=\\sqrt{x-5},\\text{ }[\/latex]find[latex]\\text{ }f+g,\\text{ }f-g,\\text{ }fg,\\text{ }[\/latex]and[latex]\\text{ }\\frac{f}{g}.\\text{ }[\/latex]Determine the domain for each function in interval notation.<\/p>\n<\/div>\n

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                        Show Solution<\/span><\/p>\n
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                        [latex]\\left(f+g\\right)\\left(x\\right)=3{x}^{2}+\\sqrt{x-5},\\text{ }[\/latex]domain:[latex]\\text{ }\\left[5,\\infty \\right)[\/latex]<\/p>\n

                        [latex]\\left(f-g\\right)\\left(x\\right)=3{x}^{2}-\\sqrt{x-5},\\text{ }[\/latex]domain:[latex]\\text{ }\\left[5,\\infty \\right)[\/latex]<\/p>\n

                        [latex]\\left(fg\\right)\\left(x\\right)=3{x}^{2}\\sqrt{x-5},\\text{ }[\/latex]domain:[latex]\\text{ }\\left[5,\\infty \\right)[\/latex]<\/p>\n

                        [latex]\\left(\\frac{f}{g}\\right)\\left(x\\right)=\\frac{3{x}^{2}}{\\sqrt{x-5}},\\text{ }[\/latex]domain:[latex]\\text{ }\\left(5,\\infty \\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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                        10. Given[latex]\\text{ }f\\left(x\\right)=\\sqrt{x}\\text{ }[\/latex]and[latex]\\text{ }g\\left(x\\right)=|x-3|,\\text{ }[\/latex]find[latex]\\text{ }\\frac{g}{f}.\\text{ }[\/latex]Determine the domain of the function in interval notation.<\/p>\n<\/div>\n<\/div>\n

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                        11. Given[latex]\\text{ }f\\left(x\\right)=2{x}^{2}+1\\text{ }[\/latex]and[latex]\\text{ }g\\left(x\\right)=3x-5,\\text{ }[\/latex]find the following:<\/p>\n

                          \n
                        1. [latex]f\\left(g\\left(2\\right)\\right)[\/latex]<\/li>\n
                        2. [latex]f\\left(g\\left(x\\right)\\right)[\/latex]<\/li>\n
                        3. [latex]g\\left(f\\left(x\\right)\\right)[\/latex]<\/li>\n
                        4. [latex]\\left(g\\circ g\\right)\\left(x\\right)[\/latex]<\/li>\n
                        5. [latex]\\left(f\\circ f\\right)\\left(-2\\right)[\/latex]<\/li>\n<\/ol>\n<\/div>\n
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                          Show Solution<\/span><\/p>\n
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                          a. 3; b.[latex]\\text{ }f\\left(g\\left(x\\right)\\right)=2{\\left(3x-5\\right)}^{2}+1;\\text{ }[\/latex]c.[latex]\\text{ }f\\left(g\\left(x\\right)\\right)=6{x}^{2}-2;\\text{ }[\/latex]d.[latex]\\text{ }\\left(g\\circ g\\right)\\left(x\\right)=3\\left(3x-5\\right)-5=9x-20;\\text{ }[\/latex]e.[latex]\\text{ }\\left(f\\circ f\\right)\\left(-2\\right)=163[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                          For the following exercises, use each pair of functions to find[latex]\\text{ }f\\left(g\\left(x\\right)\\right)\\text{ }[\/latex]and[latex]\\text{ }g\\left(f\\left(x\\right)\\right).\\text{ }[\/latex]Simplify your answers.<\/p>\n

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                          12. [latex]f\\left(x\\right)={x}^{2}+1,\\text{ }g\\left(x\\right)=\\sqrt{x+2}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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                          13. [latex]f\\left(x\\right)=\\sqrt{x}+2,\\text{ }g\\left(x\\right)={x}^{2}+3[\/latex]<\/p>\n<\/div>\n

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                          Show Solution<\/span><\/p>\n
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                          [latex]f\\left(g\\left(x\\right)\\right)=\\sqrt{{x}^{2}+3}+2,\\text{ }g\\left(f\\left(x\\right)\\right)=x+4\\sqrt{x}+7[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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                          14. [latex]f\\left(x\\right)=|x|,\\text{ }g\\left(x\\right)=5x+1[\/latex]<\/p>\n<\/div>\n<\/div>\n

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                          15. [latex]f\\left(x\\right)=\\sqrt[3]{x},\\text{ }g\\left(x\\right)=\\frac{x+1}{{x}^{3}}[\/latex]<\/p>\n<\/div>\n

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                          Show Solution<\/span><\/p>\n
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                          [latex]f\\left(g\\left(x\\right)\\right)=\\sqrt[3]{\\frac{x+1}{{x}^{3}}}=\\frac{\\sqrt[3]{x+1}}{x},\\text{ }g\\left(f\\left(x\\right)\\right)=\\frac{\\sqrt[3]{x}+1}{x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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                          16. [latex]f\\left(x\\right)=\\frac{1}{x-6},\\text{ }g\\left(x\\right)=\\frac{7}{x}+6[\/latex]<\/p>\n<\/div>\n<\/div>\n

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                          17. [latex]f\\left(x\\right)=\\frac{1}{x-4},\\text{ }g\\left(x\\right)=\\frac{2}{x}+4[\/latex]<\/p>\n<\/div>\n

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                          Show Solution<\/span><\/p>\n
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                          [latex]\\left(f\\circ g\\right)\\left(x\\right)=\\frac{1}{\\frac{2}{x}+4-4}=\\frac{x}{2},\\text{ }\\left(g\\circ f\\right)\\left(x\\right)=2x-4[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                          For the following exercises, use each set of functions to find[latex]\\text{ }f\\left(g\\left(h\\left(x\\right)\\right)\\right).\\text{ }[\/latex]Simplify your answers.<\/p>\n

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                          18. [latex]f\\left(x\\right)={x}^{4}+6,\\text{ }[\/latex][latex]g\\left(x\\right)=x-6,\\text{ }[\/latex]and[latex]\\text{ }h\\left(x\\right)=\\sqrt{x}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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                          19. [latex]f\\left(x\\right)={x}^{2}+1,\\text{ }[\/latex][latex]g\\left(x\\right)=\\frac{1}{x},\\text{ }[\/latex]and[latex]\\text{ }h\\left(x\\right)=x+3[\/latex]<\/p>\n<\/div>\n

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                          Show Solution<\/span><\/p>\n
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                          [latex]f\\left(g\\left(h\\left(x\\right)\\right)\\right)={\\left(\\frac{1}{x+3}\\right)}^{2}+1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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                          20. Given[latex]\\text{ }f\\left(x\\right)=\\frac{1}{x}\\text{ }[\/latex]and[latex]\\text{ }g\\left(x\\right)=x-3,\\text{ }[\/latex]find the following:<\/p>\n

                            \n
                          1. [latex]\\left(f\\circ g\\right)\\left(x\\right)[\/latex]<\/li>\n
                          2. the domain of[latex]\\text{ }\\left(f\\circ g\\right)\\left(x\\right)\\text{ }[\/latex]in interval notation<\/li>\n
                          3. [latex]\\left(g\\circ f\\right)\\left(x\\right)[\/latex]<\/li>\n
                          4. the domain of[latex]\\text{ }\\left(g\\circ f\\right)\\left(x\\right)\\text{ }[\/latex]<\/li>\n
                          5. [latex]\\left(\\frac{f}{g}\\right)x[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n
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                            \n

                            21. Given[latex]\\text{ }f\\left(x\\right)=\\sqrt{2-4x}\\text{ }[\/latex]and[latex]\\text{ }g\\left(x\\right)=-\\frac{3}{x},\\text{ }[\/latex]find the following:<\/p>\n

                              \n
                            1. [latex]\\left(g\\circ f\\right)\\left(x\\right)[\/latex]<\/li>\n
                            2. the domain of[latex]\\text{ }\\left(g\\circ f\\right)\\left(x\\right)\\text{ }[\/latex]in interval notation<\/li>\n<\/ol>\n<\/div>\n
                              \n
                              Show Solution<\/span><\/p>\n
                              \n

                              a.[latex]\\text{ }\\left(g\\circ f\\right)\\left(x\\right)=-\\frac{3}{\\sqrt{2-4x}};\\text{ }[\/latex]b.[latex]\\text{ }\\left(-\\infty ,\\frac{1}{2}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                              \n
                              \n

                              22. Given the functions[latex]\\text{ }f\\left(x\\right)=\\frac{1-x}{x}\\text{ }\\text{and}\\text{ }g\\left(x\\right)=\\frac{1}{1+{x}^{2}},\\text{ }[\/latex] find the following:<\/p>\n

                                \n
                              1. [latex]\\left(g\\circ f\\right)\\left(x\\right)[\/latex]<\/li>\n
                              2. [latex]\\left(g\\circ f\\right)\\left(\\text{2}\\right)[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n
                                \n
                                \n

                                23. Given functions[latex]\\text{ }p\\left(x\\right)=\\frac{1}{\\sqrt{x}}\\text{ }[\/latex]and[latex]\\text{ }m\\left(x\\right)={x}^{2}-4,\\text{ }[\/latex]state the domain of each of the following functions using interval notation:<\/p>\n

                                  \n
                                1. [latex]\\frac{p\\left(x\\right)}{m\\left(x\\right)}[\/latex]<\/li>\n
                                2. [latex]p\\left(m\\left(x\\right)\\right)[\/latex]<\/li>\n
                                3. [latex]m\\left(p\\left(x\\right)\\right)[\/latex]<\/li>\n<\/ol>\n<\/div>\n
                                  \n
                                  Show Solution<\/span><\/p>\n
                                  \n

                                  a.[latex]\\text{ }\\left(0,2\\right)\\cup \\left(2,\\infty \\right);\\text{ }[\/latex]b.[latex]\\text{ }\\left(-\\infty ,-2\\right)\\cup \\left(2,\\infty \\right);\\text{ }[\/latex]c.[latex]\\text{ }\\left(0,\\infty \\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                                  \n
                                  \n

                                  24. Given functions[latex]\\text{ }q\\left(x\\right)=\\frac{1}{\\sqrt{x}}\\text{ }[\/latex]and[latex]\\text{ }h\\left(x\\right)={x}^{2}-9,\\text{ }[\/latex]state the domain of each of the following functions using interval notation.<\/p>\n

                                    \n
                                  1. [latex]\\frac{q\\left(x\\right)}{h\\left(x\\right)}[\/latex]<\/li>\n
                                  2. [latex]q\\left(h\\left(x\\right)\\right)[\/latex]<\/li>\n
                                  3. [latex]h\\left(q\\left(x\\right)\\right)[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n
                                    \n
                                    \n

                                    25. For[latex]\\text{ }f\\left(x\\right)=\\frac{1}{x}\\text{ }[\/latex]and[latex]\\text{ }g\\left(x\\right)=\\sqrt{x-1},\\text{ }[\/latex]write the domain of[latex]\\text{ }\\left(f\\circ g\\right)\\left(x\\right)\\text{ }[\/latex]in interval notation.<\/p>\n<\/div>\n

                                    \n
                                    Show Solution<\/span><\/p>\n
                                    \n

                                    [latex]\\left(1,\\infty \\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                                    For the following exercises, find functions[latex]\\text{ }f\\left(x\\right)\\text{ }[\/latex]and[latex]\\text{ }g\\left(x\\right)\\text{ }[\/latex]so the given function can be expressed as[latex]\\text{ }h\\left(x\\right)=f\\left(g\\left(x\\right)\\right).[\/latex]<\/p>\n

                                    \n
                                    \n

                                    26. [latex]h\\left(x\\right)={\\left(x+2\\right)}^{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n

                                    \n
                                    \n

                                    27. [latex]h\\left(x\\right)={\\left(x-5\\right)}^{3}[\/latex]<\/p>\n<\/div>\n

                                    \n
                                    Show Solution<\/span><\/p>\n
                                    \n

                                    sample:[latex]\\begin{array}{l}f\\left(x\\right)={x}^{3}\\\\ g\\left(x\\right)=x-5\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                                    \n
                                    \n

                                    28. [latex]h\\left(x\\right)=\\frac{3}{x-5}[\/latex]<\/p>\n<\/div>\n<\/div>\n

                                    \n
                                    \n

                                    29. [latex]h\\left(x\\right)=\\frac{4}{{\\left(x+2\\right)}^{2}}[\/latex]<\/p>\n<\/div>\n

                                    \n
                                    Show Solution<\/span><\/p>\n
                                    \n

                                    sample:[latex]\\begin{array}{l}f\\left(x\\right)=\\frac{4}{x}\\hfill \\\\ g\\left(x\\right)={\\left(x+2\\right)}^{2}\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                                    \n
                                    \n

                                    30. [latex]h\\left(x\\right)=4+\\sqrt[3]{x}[\/latex]<\/p>\n<\/div>\n<\/div>\n

                                    \n
                                    \n

                                    31. [latex]h\\left(x\\right)=\\sqrt[3]{\\frac{1}{2x-3}}[\/latex]<\/p>\n<\/div>\n

                                    \n
                                    Show Solution<\/span><\/p>\n
                                    \n

                                    sample:[latex]\\begin{array}{l}f\\left(x\\right)=\\sqrt[3]{x}\\\\ g\\left(x\\right)=\\frac{1}{2x-3}\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                                    \n
                                    \n

                                    32. [latex]h\\left(x\\right)=\\frac{1}{{\\left(3{x}^{2}-4\\right)}^{-3}}[\/latex]<\/p>\n<\/div>\n<\/div>\n

                                    \n
                                    \n

                                    33.[latex]h\\left(x\\right)=\\sqrt[4]{\\frac{3x-2}{x+5}}[\/latex]<\/p>\n<\/div>\n

                                    \n
                                    Show Solution<\/span><\/p>\n
                                    \n

                                    sample:[latex]\\begin{array}{l}f\\left(x\\right)=\\sqrt[4]{x}\\\\ g\\left(x\\right)=\\frac{3x-2}{x+5}\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                                    \n
                                    \n

                                    34. [latex]h\\left(x\\right)={\\left(\\frac{8+{x}^{3}}{8-{x}^{3}}\\right)}^{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n

                                    \n
                                    \n

                                    35. [latex]h\\left(x\\right)=\\sqrt{2x+6}[\/latex]<\/p>\n<\/div>\n

                                    \n
                                    Show Solution<\/span><\/p>\n
                                    \n

                                    sample: [latex]f\\left(x\\right)=\\sqrt{x}[\/latex]<\/p>\n

                                    <\/div>\n<\/div>\n<\/div>\n
                                    <\/div>\n
                                    36. [latex]g\\left(x\\right)=2x+6[\/latex]<\/div>\n<\/div>\n<\/div>\n
                                    \n
                                    \n

                                    37. [latex]h\\left(x\\right)={\\left(5x-1\\right)}^{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n

                                    \n
                                    \n

                                    38. [latex]h\\left(x\\right)=\\sqrt[3]{x-1}[\/latex]<\/p>\n<\/div>\n

                                    \n
                                    Show Solution<\/span><\/p>\n
                                    \n

                                    sample: [latex]f\\left(x\\right)=\\sqrt[3]{x}[\/latex]<\/p>\n

                                    <\/div>\n<\/div>\n<\/div>\n

                                    39. [latex]g\\left(x\\right)=\\left(x-1\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

                                    \n
                                    \n

                                    40. [latex]h\\left(x\\right)=|{x}^{2}+7|[\/latex]<\/p>\n<\/div>\n<\/div>\n

                                    \n
                                    \n

                                    41. [latex]h\\left(x\\right)=\\frac{1}{{\\left(x-2\\right)}^{3}}[\/latex]<\/p>\n<\/div>\n

                                    \n
                                    Show Solution<\/span><\/p>\n
                                    \n

                                    sample: [latex]f\\left(x\\right)={x}^{3}[\/latex]<\/p>\n

                                    <\/div>\n<\/div>\n<\/div>\n

                                    42. [latex]g\\left(x\\right)=\\frac{1}{x-2}[\/latex]<\/p>\n<\/div>\n<\/div>\n

                                    \n
                                    \n

                                    43. [latex]h\\left(x\\right)={\\left(\\frac{1}{2x-3}\\right)}^{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n

                                    \n
                                    \n

                                    44. [latex]h\\left(x\\right)=\\sqrt{\\frac{2x-1}{3x+4}}[\/latex]<\/p>\n<\/div>\n

                                    \n
                                    Show Solution<\/span><\/p>\n
                                    \n

                                    sample: [latex]f\\left(x\\right)=\\sqrt{x}[\/latex]<\/p>\n

                                    <\/div>\n<\/div>\n<\/div>\n

                                    45. [latex]g\\left(x\\right)=\\frac{2x-1}{3x+4}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n

                                    \n

                                    Graphical<\/h4>\n

                                    For the following exercises, use the graphs of[latex]\\text{ }f,[\/latex]shown in (Figure)<\/a>, and[latex]\\text{ }g,[\/latex]shown in (Figure)<\/a>, to evaluate the expressions.<\/p>\n

                                    \"graph<\/span><\/div>\n
                                    \u00a0<\/span><\/div>\n
                                    \"\"<\/div>\n
                                    \n
                                    \n

                                    46. [latex]f\\left(g\\left(3\\right)\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

                                    \n
                                    \n

                                    47. [latex]f\\left(g\\left(1\\right)\\right)[\/latex]<\/p>\n<\/div>\n

                                    \n
                                    Show Solution<\/span><\/p>\n
                                    \n

                                    2<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                                    \n
                                    \n

                                    48. [latex]g\\left(f\\left(1\\right)\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

                                    \n
                                    \n

                                    49. [latex]g\\left(f\\left(0\\right)\\right)[\/latex]<\/p>\n<\/div>\n

                                    \n
                                    Show Solution<\/span><\/p>\n
                                    \n

                                    5<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                                    \n
                                    \n

                                    50. [latex]f\\left(f\\left(5\\right)\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

                                    \n
                                    \n

                                    51. [latex]f\\left(f\\left(4\\right)\\right)[\/latex]<\/p>\n<\/div>\n

                                    \n
                                    Show Solution<\/span><\/p>\n
                                    \n

                                    4<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                                    \n
                                    \n

                                    52. [latex]g\\left(g\\left(2\\right)\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

                                    \n
                                    \n

                                    53. [latex]g\\left(g\\left(0\\right)\\right)[\/latex]<\/p>\n<\/div>\n

                                    \n
                                    Show Solution<\/span><\/p>\n
                                    \n

                                    0<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                                    For the following exercises, use graphs of[latex]\\text{ }f\\left(x\\right),[\/latex]shown in (Figure)<\/a>,[latex]\\text{ }g\\left(x\\right),[\/latex]shown in (Figure)<\/a>, and[latex]\\text{ }h\\left(x\\right),[\/latex]shown in (Figure)<\/a>, to evaluate the expressions.<\/p>\n

                                    \"graph<\/p>\n

                                    \"\"<\/p>\n

                                    \"\"<\/p>\n

                                    <\/div>\n
                                    <\/div>\n
                                    \n
                                    \n

                                    54. [latex]g\\left(f\\left(1\\right)\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

                                    \n
                                    \n

                                    55. [latex]g\\left(f\\left(2\\right)\\right)[\/latex]<\/p>\n<\/div>\n

                                    \n
                                    Show Solution<\/span><\/p>\n
                                    \n

                                    2<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                                    \n
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                                    56. [latex]f\\left(g\\left(4\\right)\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

                                    \n
                                    \n

                                    57. [latex]f\\left(g\\left(1\\right)\\right)[\/latex]<\/p>\n<\/div>\n

                                    \n
                                    Show Solution<\/span><\/p>\n
                                    \n

                                    1<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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                                    58. [latex]f\\left(h\\left(2\\right)\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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                                    \n

                                    59. [latex]h\\left(f\\left(2\\right)\\right)[\/latex]<\/p>\n<\/div>\n

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                                    Show Solution<\/span><\/p>\n
                                    \n

                                    4<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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                                    60. [latex]f\\left(g\\left(h\\left(4\\right)\\right)\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

                                    \n
                                    \n

                                    61. [latex]f\\left(g\\left(f\\left(-2\\right)\\right)\\right)[\/latex]<\/p>\n<\/div>\n

                                    \n
                                    Show Solution<\/span><\/p>\n
                                    \n

                                    4<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                                    \n

                                    Numeric<\/h4>\n

                                    For the following exercises, use the function values for[latex]\\text{ }f\\text{ and }g\\text{ }[\/latex]shown in (Figure)<\/a> to evaluate each expression.<\/p>\n\n\n\n\n\n\n\n\n\n\n\n\n\n
                                    [latex]x[\/latex]<\/strong><\/td>\n[latex]f\\left(x\\right)[\/latex]<\/strong><\/td>\n[latex]g\\left(x\\right)[\/latex]<\/strong><\/td>\n<\/tr>\n
                                    0<\/td>\n7<\/td>\n9<\/td>\n<\/tr>\n
                                    1<\/td>\n6<\/td>\n5<\/td>\n<\/tr>\n
                                    2<\/td>\n5<\/td>\n6<\/td>\n<\/tr>\n
                                    3<\/td>\n8<\/td>\n2<\/td>\n<\/tr>\n
                                    4<\/td>\n4<\/td>\n1<\/td>\n<\/tr>\n
                                    5<\/td>\n0<\/td>\n8<\/td>\n<\/tr>\n
                                    6<\/td>\n2<\/td>\n7<\/td>\n<\/tr>\n
                                    7<\/td>\n1<\/td>\n3<\/td>\n<\/tr>\n
                                    8<\/td>\n9<\/td>\n4<\/td>\n<\/tr>\n
                                    9<\/td>\n3<\/td>\n0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
                                    \n
                                    \n

                                    62. [latex]f\\left(g\\left(8\\right)\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

                                    \n
                                    \n

                                    63. [latex]f\\left(g\\left(5\\right)\\right)[\/latex]<\/p>\n<\/div>\n

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                                    Show Solution<\/span><\/p>\n
                                    \n

                                    9<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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                                    64. [latex]g\\left(f\\left(5\\right)\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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                                    65. [latex]g\\left(f\\left(3\\right)\\right)[\/latex]<\/p>\n<\/div>\n

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                                    \n

                                    4<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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                                    66. [latex]f\\left(f\\left(4\\right)\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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                                    \n

                                    67. [latex]f\\left(f\\left(1\\right)\\right)[\/latex]<\/p>\n<\/div>\n

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                                    Show Solution<\/span><\/p>\n
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                                    2<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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                                    68. [latex]g\\left(g\\left(2\\right)\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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                                    \n

                                    69. [latex]g\\left(g\\left(6\\right)\\right)[\/latex]<\/p>\n<\/div>\n

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                                    Show Solution<\/span><\/p>\n
                                    \n

                                    3<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                                    For the following exercises, use the function values for[latex]\\text{ }f\\text{ and }g\\text{ }[\/latex]shown in (Figure)<\/a> to evaluate the expressions.<\/p>\n\n\n\n\n\n\n\n\n\n\n
                                    [latex]x[\/latex]<\/strong><\/td>\n[latex]f\\left(x\\right)[\/latex]<\/strong><\/td>\n[latex]g\\left(x\\right)[\/latex]<\/strong><\/td>\n<\/tr>\n
                                    -3<\/td>\n11<\/td>\n-8<\/td>\n<\/tr>\n
                                    -2<\/td>\n9<\/td>\n-3<\/td>\n<\/tr>\n
                                    -1<\/td>\n7<\/td>\n0<\/td>\n<\/tr>\n
                                    0<\/td>\n5<\/td>\n1<\/td>\n<\/tr>\n
                                    1<\/td>\n3<\/td>\n0<\/td>\n<\/tr>\n
                                    2<\/td>\n1<\/td>\n-3<\/td>\n<\/tr>\n
                                    3<\/td>\n-1<\/td>\n-8<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
                                    \n
                                    \n

                                    70. [latex]\\left(f\\circ g\\right)\\left(1\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

                                    \n
                                    \n

                                    71. [latex]\\left(f\\circ g\\right)\\left(2\\right)[\/latex]<\/p>\n<\/div>\n

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                                    \n

                                    11<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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                                    72. [latex]\\left(g\\circ f\\right)\\left(2\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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                                    \n

                                    73. [latex]\\left(g\\circ f\\right)\\left(3\\right)[\/latex]<\/p>\n<\/div>\n

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                                    \n

                                    0<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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                                    74. [latex]\\left(g\\circ g\\right)\\left(1\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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                                    75. [latex]\\left(f\\circ f\\right)\\left(3\\right)[\/latex]<\/p>\n<\/div>\n

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                                    Show Solution<\/span><\/p>\n
                                    \n

                                    7<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                                    For the following exercises, use each pair of functions to find[latex]\\text{ }f\\left(g\\left(0\\right)\\right)\\text{ }[\/latex]and[latex]\\text{ }g\\left(f\\left(0\\right)\\right).[\/latex]<\/p>\n

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                                    76. [latex]f\\left(x\\right)=4x+8,\\text{ }g\\left(x\\right)=7-{x}^{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n

                                    \n
                                    \n

                                    77. [latex]f\\left(x\\right)=5x+7,\\text{ }g\\left(x\\right)=4-2{x}^{2}[\/latex]<\/p>\n<\/div>\n

                                    \n
                                    Show Solution<\/span><\/p>\n
                                    \n

                                    [latex]f\\left(g\\left(0\\right)\\right)=27,\\text{ }g\\left(f\\left(0\\right)\\right)=-94[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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                                    78. [latex]f\\left(x\\right)=\\sqrt{x+4},\\text{ }g\\left(x\\right)=12-{x}^{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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                                    79. [latex]f\\left(x\\right)=\\frac{1}{x+2},\\text{ }g\\left(x\\right)=4x+3[\/latex]<\/p>\n<\/div>\n

                                    \n
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                                    \n

                                    [latex]f\\left(g\\left(0\\right)\\right)=\\frac{1}{5},\\text{ }g\\left(f\\left(0\\right)\\right)=5[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                                    For the following exercises, use the functions[latex]\\text{ }f\\left(x\\right)=2{x}^{2}+1\\text{ }[\/latex]and[latex]\\text{ }g\\left(x\\right)=3x+5\\text{ }[\/latex] to evaluate or find the composite function as indicated.<\/p>\n

                                    \n
                                    \n

                                    80. [latex]f\\left(g\\left(2\\right)\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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                                    \n

                                    81. [latex]f\\left(g\\left(x\\right)\\right)[\/latex]<\/p>\n<\/div>\n

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                                    Show Solution<\/span><\/p>\n
                                    \n

                                    [latex]18{x}^{2}+60x+51[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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                                    82. [latex]g\\left(f\\left(-3\\right)\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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                                    83. [latex]\\left(g\\circ g\\right)\\left(x\\right)[\/latex]<\/p>\n<\/div>\n

                                    \n
                                    Show Solution<\/span><\/p>\n
                                    \n

                                    [latex]g\\circ g\\left(x\\right)=9x+20[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                                    \n

                                    Extensions<\/h4>\n

                                    For the following exercises, use[latex]\\text{ }f\\left(x\\right)={x}^{3}+1\\text{ }[\/latex]and[latex]\\text{ }g\\left(x\\right)=\\sqrt[3]{x-1}.[\/latex]<\/p>\n

                                    \n
                                    \n

                                    84. Find[latex]\\text{ }\\left(f\\circ g\\right)\\left(x\\right)\\text{ }[\/latex]and[latex]\\text{ }\\left(g\\circ f\\right)\\left(x\\right).\\text{ }[\/latex]Compare the two answers.<\/p>\n<\/div>\n<\/div>\n

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                                    85. Find[latex]\\text{ }\\left(f\\circ g\\right)\\left(2\\right)\\text{ }[\/latex]and[latex]\\text{ }\\left(g\\circ f\\right)\\left(2\\right).[\/latex]<\/p>\n<\/div>\n

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                                    2<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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                                    86. What is the domain of[latex]\\text{ }\\left(g\\circ f\\right)\\left(x\\right)?[\/latex]<\/p>\n<\/div>\n<\/div>\n

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                                    87. What is the domain of[latex]\\text{ }\\left(f\\circ g\\right)\\left(x\\right)?[\/latex]<\/p>\n<\/div>\n

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                                    [latex]\\left(-\\infty ,\\infty \\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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                                    88. Let[latex]\\text{ }f\\left(x\\right)=\\frac{1}{x}.[\/latex]<\/p>\n

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                                    1. Find[latex]\\text{ }\\left(f\\circ f\\right)\\left(x\\right).[\/latex]<\/li>\n
                                    2. Is[latex]\\text{ }\\left(f\\circ f\\right)\\left(x\\right)\\text{ }[\/latex]for any function[latex]\\text{ }f\\text{ }[\/latex]the same result as the answer to part (a) for any function? Explain.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n

                                      For the following exercises, let[latex]\\text{ }F\\left(x\\right)={\\left(x+1\\right)}^{5},\\text{ }[\/latex][latex]f\\left(x\\right)={x}^{5},\\text{ }[\/latex]and[latex]\\text{ }g\\left(x\\right)=x+1.[\/latex]<\/p>\n

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                                      89. True or False:[latex]\\text{ }\\left(g\\circ f\\right)\\left(x\\right)=F\\left(x\\right).[\/latex]<\/p>\n<\/div>\n

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                                      False<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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                                      90. True or False:[latex]\\text{ }\\left(f\\circ g\\right)\\left(x\\right)=F\\left(x\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n

                                      For the following exercises, find the composition when[latex]\\text{ }f\\left(x\\right)={x}^{2}+2\\text{ }[\/latex]for all[latex]\\text{ }x\\ge 0\\text{ }[\/latex]and[latex]\\text{ }g\\left(x\\right)=\\sqrt{x-2}.[\/latex]<\/p>\n

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                                      91. [latex]\\left(f\\circ g\\right)\\left(6\\right);\\text{ }\\left(g\\circ f\\right)\\left(6\\right)[\/latex]<\/p>\n<\/div>\n

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                                      [latex]\\left(f\\circ g\\right)\\left(6\\right)=6[\/latex]; [latex]\\text{ }\\left(g\\circ f\\right)\\left(6\\right)=6[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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                                      92. [latex]\\left(g\\circ f\\right)\\left(a\\right);\\text{ }\\left(f\\circ g\\right)\\left(a\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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                                      93. [latex]\\left(f\\circ g\\right)\\left(11\\right);\\text{ }\\left(g\\circ f\\right)\\left(11\\right)[\/latex]<\/p>\n<\/div>\n

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                                      [latex]\\left(f\\circ g\\right)\\left(11\\right)=11\\text{ },\\text{ }\\left(g\\circ f\\right)\\left(11\\right)=11[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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                                      Real-World Applications<\/h4>\n
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                                      94. The function[latex]\\text{ }D\\left(p\\right)\\text{ }[\/latex]gives the number of items that will be demanded when the price is[latex]\\text{ }p.\\text{ }[\/latex]The production cost[latex]\\text{ }C\\left(x\\right)\\text{ }[\/latex]is the cost of producing[latex]\\text{ }x\\text{ }[\/latex]items. To determine the cost of production when the price is $6, you would do which of the following?<\/p>\n

                                        \n
                                      1. Evaluate[latex]\\text{ }D\\left(C\\left(6\\right)\\right).[\/latex]<\/li>\n
                                      2. Evaluate[latex]\\text{ }C\\left(D\\left(6\\right)\\right).[\/latex]<\/li>\n
                                      3. Solve[latex]\\text{ }D\\left(C\\left(x\\right)\\right)=6.[\/latex]<\/li>\n
                                      4. Solve[latex]\\text{ }C\\left(D\\left(p\\right)\\right)=6.[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n
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                                        95. The function[latex]\\text{ }A\\left(d\\right)\\text{ }[\/latex]gives the pain level on a scale of 0 to 10 experienced by a patient with[latex]\\text{ }d\\text{ }[\/latex]milligrams of a pain-reducing drug in her system. The milligrams of the drug in the patient\u2019s system after[latex]\\text{ }t\\text{ }[\/latex]minutes is modeled by[latex]\\text{ }m\\left(t\\right).\\text{ }[\/latex]Which of the following would you do in order to determine when the patient will be at a pain level of 4?<\/p>\n

                                          \n
                                        1. Evaluate[latex]\\text{ }A\\left(m\\left(4\\right)\\right).[\/latex]<\/li>\n
                                        2. Evaluate[latex]\\text{ }m\\left(A\\left(4\\right)\\right).[\/latex]<\/li>\n
                                        3. Solve[latex]\\text{ }A\\left(m\\left(t\\right)\\right)=4.[\/latex]<\/li>\n
                                        4. Solve[latex]\\text{ }m\\left(A\\left(d\\right)\\right)=4.[\/latex]<\/li>\n<\/ol>\n<\/div>\n
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                                          c<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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                                          96. A store offers customers a 30% discount on the price[latex]\\text{ }x\\text{ }[\/latex]of selected items. Then, the store takes off an additional 15% at the cash register. Write a price function[latex]\\text{ }P\\left(x\\right)\\text{ }[\/latex]that computes the final price of the item in terms of the original price[latex]\\text{ }x.\\text{ }[\/latex](Hint: Use function composition to find your answer.)<\/p>\n<\/div>\n<\/div>\n

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                                          97. A rain drop hitting a lake makes a circular ripple. If the radius, in inches, grows as a function of time in minutes according to[latex]\\text{ }r\\left(t\\right)=25\\sqrt{t+2},\\text{ }[\/latex]find the area of the ripple as a function of time. Find the area of the ripple at[latex]\\text{ }t=2.[\/latex]<\/p>\n<\/div>\n

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                                          [latex]A\\left(t\\right)=\\pi {\\left(25\\sqrt{t+2}\\right)}^{2}\\text{ }[\/latex]and[latex]\\text{ }A\\left(2\\right)=\\pi {\\left(25\\sqrt{4}\\right)}^{2}=2500\\pi[\/latex] square inches<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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                                          98. A forest fire leaves behind an area of grass burned in an expanding circular pattern. If the radius of the circle of burning grass is increasing with time according to the formula[latex]\\text{ }r\\left(t\\right)=2t+1,\\text{ }[\/latex]express the area burned as a function of time,[latex]\\text{ }t\\text{ }[\/latex](minutes).<\/p>\n<\/div>\n<\/div>\n

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                                          99. Use the function you found in the previous exercise to find the total area burned after 5 minutes.<\/p>\n<\/div>\n

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                                          Show Solution<\/span><\/p>\n
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                                          [latex]A\\left(5\\right)=\\pi {\\left(2\\left(5\\right)+1\\right)}^{2}=121\\pi \\text{ }[\/latex]square units<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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                                          100. The radius[latex]\\text{ }r,\\text{ }[\/latex]in inches, of a spherical balloon is related to the volume,[latex]\\text{ }V,\\text{ }[\/latex]by[latex]\\text{ }r\\left(V\\right)=\\sqrt[3]{\\frac{3V}{4\\pi }}.\\text{ }[\/latex]Air is pumped into the balloon, so the volume after[latex]\\text{ }t\\text{ }[\/latex]seconds is given by[latex]\\text{ }V\\left(t\\right)=10+20t.[\/latex]<\/p>\n

                                            \n
                                          1. Find the composite function[latex]\\text{ }r\\left(V\\left(t\\right)\\right).[\/latex]<\/li>\n
                                          2. Find the exact<\/em> time when the radius reaches 10 inches.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n
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                                            101. The number of bacteria in a refrigerated food product is given by [latex]N\\left(T\\right)=23{T}^{2}-56T+1,\\text{ }[\/latex][latex]3\n

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                                            1. Find the composite function[latex]\\text{ }N\\left(T\\left(t\\right)\\right).[\/latex]<\/li>\n
                                            2. Find the time (round to two decimal places) when the bacteria count reaches 6752.<\/li>\n<\/ol>\n<\/div>\n
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                                              Show Solution<\/span><\/p>\n
                                              \n

                                              a.[latex]\\text{ }N\\left(T\\left(t\\right)\\right)=23{\\left(5t+1.5\\right)}^{2}-56\\left(5t+1.5\\right)+1;\\text{ }[\/latex] b. 3.38 hours<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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