{"id":2355,"date":"2019-05-13T12:42:57","date_gmt":"2019-05-13T12:42:57","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/?post_type=chapter&p=2355"},"modified":"2019-05-13T12:47:38","modified_gmt":"2019-05-13T12:47:38","slug":"2-1-section-exercises","status":"web-only","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/chapter\/2-1-section-exercises\/","title":{"raw":"2.1 Section Exercises","rendered":"2.1 Section Exercises"},"content":{"raw":"
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2.1 Section Exercises<\/h3>\r\n
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Verbal<\/h4>\r\n
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1. Describe why the horizontal line test is an effective way to determine whether a function is one-to-one?<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137517264\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137517264\"]\r\n

Each output of a function must have exactly one output for the function to be one-to-one. If any horizontal line crosses the graph of a function more than once, that means that[latex]\\text{ }y[\/latex]-values repeat and the function is not one-to-one. If no horizontal line crosses the graph of the function more than once, then no[latex]\\text{ }y[\/latex]-values repeat and the function is one-to-one.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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2. Why do we restrict the domain of the function[latex]\\text{ }f\\left(x\\right)={x}^{2}\\text{ }[\/latex]to find the function\u2019s inverse?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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3. Can a function be its own inverse? Explain.<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137553896\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137553896\"]\r\n

Yes. For example,[latex]\\text{ }f\\left(x\\right)=\\frac{1}{x}\\text{ }[\/latex]is its own inverse.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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4. Are one-to-one functions either always increasing or always decreasing? Why or why not?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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5. How do you find the inverse of a function algebraically?<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137785042\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137785042\"]\r\n

Given a function[latex]\\text{ }y=f\\left(x\\right),\\text{ }[\/latex]solve for[latex]\\text{ }x\\text{ }[\/latex]in terms of[latex]\\text{ }y.\\text{ }[\/latex]The expression for[latex]\\text{ }x\\text{ }[\/latex]is the inverse,[latex]\\text{ }{f}^{-1}\\left(y\\right).[\/latex]<\/p>\r\n[\/hidden-answer]\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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6. Show that the function[latex]\\text{ }f\\left(x\\right)=a-x\\text{ }[\/latex]is its own inverse for all real numbers[latex]\\text{ }a.\\text{ }[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, find[latex]\\text{ }{f}^{-1}\\left(x\\right)\\text{ }[\/latex]for each function.<\/p>\r\n\r\n

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7. [latex]f\\left(x\\right)=x+3[\/latex]<\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n[reveal-answer q=\"503136\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"503136\"][latex]{f}^{-1}\\left(y\\right)=y-3[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n
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8. [latex]f\\left(x\\right)=x+5[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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9. [latex]f\\left(x\\right)=2-x[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137812372\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137812372\"]\r\n

[latex]{f}^{-1}\\left(y\\right)=2-y[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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10. [latex]f\\left(x\\right)=3-x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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11. [latex]f\\left(x\\right)=\\frac{x}{x+2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135541959\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135541959\"]\r\n

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

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

For the following exercises, find a domain on which each function[latex]\\text{ }f\\text{ }[\/latex]is one-to-one and non-decreasing. Write the domain in interval notation. Then find the inverse of[latex]\\text{ }f\\text{ }[\/latex]restricted to that domain.<\/p>\r\n\r\n

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

[reveal-answer q=\"fs-id1165137849508\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137849508\"]\r\n

domain of[latex]f\\left(x\\right):\\text{ }\\left[-7,\\infty \\right);\\text{ }{f}^{-1}\\left(y\\right)=\\sqrt{y}-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)={\\left(x-6\\right)}^{2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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15. [latex]f\\left(x\\right)={x}^{2}-5[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137603366\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137603366\"]\r\n

domain of[latex]\\text{ }f\\left(x\\right):\\text{ }\\left[0,\\infty \\right);\\text{ }{f}^{-1}\\left(y\\right)=\\sqrt{y+5}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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16. Given[latex]\\text{ }f\\left(x\\right)=\\frac{x}{2+x}\\text{ }[\/latex]and[latex]\\text{ }g\\left(x\\right)=\\frac{2x}{1-x}:[\/latex]<\/p>\r\n\r\n

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  1. Find[latex]\\text{ }f\\left(g\\left(x\\right)\\right)\\text{ }[\/latex]and[latex]\\text{ }g\\left(f\\left(x\\right)\\right).[\/latex]<\/li>\r\n \t
  2. What does the answer tell us about the relationship between[latex]\\text{ }f\\left(x\\right)\\text{ }[\/latex]and[latex]\\text{ }g\\left(x\\right)?[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n
    [reveal-answer q=\"fs-id1165137730082\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137730082\"]\r\n

    a.[latex] \\text{ }f\\left(g\\left(x\\right)\\right)=x\\text{ }[\/latex]and[latex]\\text{ }g\\left(f\\left(x\\right)\\right)=x.\\text{ }[\/latex]b. This tells us that[latex]\\text{ }f\\text{ }[\/latex]and[latex]\\text{ }g\\text{ }[\/latex]are inverse functions<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

    For the following exercises, use function composition to verify that[latex]\\text{ }f\\left(x\\right)\\text{ }[\/latex]and[latex]\\text{ }g\\left(x\\right)\\text{ }[\/latex]are inverse functions.<\/p>\r\n\r\n

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

    [reveal-answer q=\"fs-id1165137883792\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137883792\"]\r\n

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

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    18. [latex]f\\left(x\\right)=-3x+5\\text{ }[\/latex]and[latex]\\text{ }g\\left(x\\right)=\\frac{x-5}{-3}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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    Graphical<\/h4>\r\n

    For the following exercises, use a graphing utility to determine whether each function is one-to-one.<\/p>\r\n\r\n

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    \r\n\r\n19. [latex]f\\left(x\\right)=\\sqrt{x}[\/latex]\r\n\r\n<\/div>\r\n
    [reveal-answer q=\"fs-id1165134113903\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134113903\"]\r\n

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

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    20. [latex]f\\left(x\\right)=\\sqrt[3]{3x+1}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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    21. [latex]f\\left(x\\right)=-5x+1[\/latex]<\/p>\r\n\r\n<\/div>\r\n

    [reveal-answer q=\"fs-id1165137433244\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137433244\"]\r\n

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

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

    For the following exercises, determine whether the graph represents a one-to-one function.<\/p>\r\n\r\n

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    23.\u00a0\"Graph<\/span><\/div>\r\n
    [reveal-answer q=\"fs-id1165137612243\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137612243\"]\r\n

    not one-to-one<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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    24.\u00a0\"Graph<\/span><\/div>\r\n<\/div>\r\n

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

    \r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"368\"]\"Graph Figure 11.<\/strong>[\/caption]\r\n\r\n<\/div>\r\n
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    25.Find[latex]\\text{ }f\\left(0\\right).[\/latex]<\/p>\r\n\r\n<\/div>\r\n

    [reveal-answer q=\"fs-id1165137653684\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137653684\"]\r\n

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

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    26. Solve[latex]\\text{ }f\\left(x\\right)=0.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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    27. Find[latex]\\text{ }{f}^{-1}\\left(0\\right).[\/latex]<\/p>\r\n\r\n<\/div>\r\n

    [reveal-answer q=\"fs-id1165137573270\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137573270\"]\r\n

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

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    28. Solve[latex]\\text{ }{f}^{-1}\\left(x\\right)=0.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

    For the following exercises, use the graph of the one-to-one function shown in (Figure)<\/a>.<\/p>\r\n\r\n

    \r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"255\"]\"Graph Figure 12.<\/strong>[\/caption]\r\n\r\n<\/div>\r\n
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    29. Sketch the graph of[latex]\\text{ }{f}^{-1}.\\text{ }[\/latex]<\/p>\r\n\r\n<\/div>\r\n

    [reveal-answer q=\"fs-id1165134113897\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134113897\"]\"Graph<\/span>[\/hidden-answer]<\/div>\r\n<\/div>\r\n
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    30. Find[latex]\\text{ }f\\left(6\\right)\\text{ and }{f}^{-1}\\left(2\\right).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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    31. If the complete graph of[latex]\\text{ }f\\text{ }[\/latex]is shown, find the domain of[latex]\\text{ }f.\\text{ }[\/latex]<\/p>\r\n\r\n<\/div>\r\n

    [reveal-answer q=\"fs-id1165134259264\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134259264\"]\r\n

    [latex]\\left[2,10\\right][\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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    32. If the complete graph of[latex]\\text{ }f\\text{ }[\/latex]is shown, find the range of[latex]\\text{ }f.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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    Numeric<\/h4>\r\n

    For the following exercises, evaluate or solve, assuming that the function[latex]\\text{ }f\\text{ }[\/latex]is one-to-one.<\/p>\r\n\r\n

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    33. If[latex]\\text{ }f\\left(6\\right)=7,\\text{ }[\/latex]find[latex]\\text{ }\\text{ }{f}^{-1}\\left(7\\right).[\/latex]<\/p>\r\n\r\n<\/div>\r\n

    [reveal-answer q=\"fs-id1165137714192\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137714192\"]\r\n

    [latex]6[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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    34. If[latex]\\text{ }f\\left(3\\right)=2,\\text{ }[\/latex]find[latex]\\text{ }{f}^{-1}\\left(2\\right).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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    35. If[latex]\\text{ }{f}^{-1}\\left(-4\\right)=-8,\\text{ }[\/latex]find[latex]\\text{ }f\\left(-8\\right).[\/latex]<\/p>\r\n\r\n<\/div>\r\n

    [reveal-answer q=\"fs-id1165137848946\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137848946\"]\r\n

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

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    36. If[latex]\\text{ }{f}^{-1}\\left(-2\\right)=-1,\\text{ }[\/latex]find[latex]\\text{ }f\\left(-1\\right).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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

    [reveal-answer q=\"fs-id1165135263655\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135263655\"]\r\n

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

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    38. Solve[latex]\\text{ }f\\left(x\\right)=3.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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    39. Find[latex]\\text{ }{f}^{-1}\\left(0\\right).[\/latex]<\/p>\r\n\r\n<\/div>\r\n

    [reveal-answer q=\"fs-id1165137422471\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137422471\"]\r\n

    [latex]\\text{ }1\\text{ }[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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    40. Solve[latex]\\text{ }{f}^{-1}\\left(x\\right)=7.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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    41. Use the tabular representation of[latex]\\text{ }f\\text{ }[\/latex]in (Figure)<\/a> to create a table for[latex]\\text{ }{f}^{-1}\\left(x\\right).[\/latex]<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n
    Table 7<\/strong><\/td>\r\n<\/tr>\r\n
    [latex]x[\/latex]<\/strong><\/td>\r\n3<\/td>\r\n6<\/td>\r\n9<\/td>\r\n13<\/td>\r\n14<\/td>\r\n<\/tr>\r\n
    [latex]f\\left(x\\right)[\/latex]<\/strong><\/td>\r\n1<\/td>\r\n4<\/td>\r\n7<\/td>\r\n12<\/td>\r\n16<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n
    [reveal-answer q=\"fs-id1165137406963\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137406963\"]\r\n\r\n\r\n\r\n\r\n
    [latex]x[\/latex]<\/td>\r\n1<\/td>\r\n4<\/td>\r\n7<\/td>\r\n12<\/td>\r\n16<\/td>\r\n<\/tr>\r\n
    [latex]{f}^{-1}\\left(x\\right)[\/latex]<\/td>\r\n3<\/td>\r\n6<\/td>\r\n9<\/td>\r\n13<\/td>\r\n14<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n
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    Technology<\/h4>\r\n

    For the following exercises, find the inverse function. Then, graph the function and its inverse.<\/p>\r\n\r\n

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

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

    [reveal-answer q=\"fs-id1165135541804\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135541804\"]\r\n

    [latex]{f}^{-1}\\left(x\\right)={\\left(1+x\\right)}^{1\/3}[\/latex]<\/p>\r\n\"Graph<\/span>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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    44. Find the inverse function of[latex]\\text{ }f\\left(x\\right)=\\frac{1}{x-1}.\\text{ }[\/latex]Use a graphing utility to find its domain and range. Write the domain and range in interval notation.<\/p>\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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    45. To convert from[latex]\\text{ }x\\text{ }[\/latex]degrees Celsius to[latex]\\text{ }y\\text{ }[\/latex]degrees Fahrenheit, we use the formula[latex]\\text{ }f\\left(x\\right)=\\frac{9}{5}x+32.\\text{ }[\/latex]Find the inverse function, if it exists, and explain its meaning.<\/p>\r\n\r\n<\/div>\r\n

    [reveal-answer q=\"fs-id1165134164967\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134164967\"]\r\n

    [latex]{f}^{-1}\\left(x\\right)=\\frac{5}{9}\\left(x-32\\right).\\text{ }[\/latex]Given the Fahrenheit temperature,[latex]\\text{ }x,\\text{ }[\/latex]this formula allows you to calculate the Celsius temperature.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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    46. The circumference[latex]\\text{ }C\\text{ }[\/latex]of a circle is a function of its radius given by[latex]\\text{ }C\\left(r\\right)=2\\pi r.\\text{ }[\/latex]Express the radius of a circle as a function of its circumference. Call this function[latex]\\text{ }r\\left(C\\right).\\text{ }[\/latex]Find[latex]\\text{ }r\\left(36\\pi \\right)\\text{ }[\/latex]and interpret its meaning.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

    \r\n
    \r\n

    47. A car travels at a constant speed of 50 miles per hour. The distance the car travels in miles is a function of time,[latex]\\text{ }t,\\text{ }[\/latex]in hours given by[latex]\\text{ }d\\left(t\\right)=50t.\\text{ }[\/latex]Find the inverse function by expressing the time of travel in terms of the distance traveled. Call this function[latex]\\text{ }t\\left(d\\right).\\text{ }[\/latex]Find[latex]\\text{ }t\\left(180\\right)\\text{ }[\/latex]and interpret its meaning.<\/p>\r\n\r\n<\/div>\r\n

    [reveal-answer q=\"fs-id1165137766909\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137766909\"]\r\n

    [latex]t\\left(d\\right)=\\frac{d}{50},\\text{ }[\/latex][latex]t\\left(180\\right)=\\frac{180}{50}.\\text{ }[\/latex]The time for the car to travel 180 miles is 3.6 hours.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>","rendered":"

    \n

    2.1 Section Exercises<\/h3>\n
    \n

    Verbal<\/h4>\n
    \n
    \n

    1. Describe why the horizontal line test is an effective way to determine whether a function is one-to-one?<\/p>\n<\/div>\n

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

    Each output of a function must have exactly one output for the function to be one-to-one. If any horizontal line crosses the graph of a function more than once, that means that[latex]\\text{ }y[\/latex]-values repeat and the function is not one-to-one. If no horizontal line crosses the graph of the function more than once, then no[latex]\\text{ }y[\/latex]-values repeat and the function is one-to-one.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

    \n
    \n

    2. Why do we restrict the domain of the function[latex]\\text{ }f\\left(x\\right)={x}^{2}\\text{ }[\/latex]to find the function\u2019s inverse?<\/p>\n<\/div>\n<\/div>\n

    \n
    \n

    3. Can a function be its own inverse? Explain.<\/p>\n<\/div>\n

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

    Yes. For example,[latex]\\text{ }f\\left(x\\right)=\\frac{1}{x}\\text{ }[\/latex]is its own inverse.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

    \n
    \n

    4. Are one-to-one functions either always increasing or always decreasing? Why or why not?<\/p>\n<\/div>\n<\/div>\n

    \n
    \n

    5. How do you find the inverse of a function algebraically?<\/p>\n<\/div>\n

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

    Given a function[latex]\\text{ }y=f\\left(x\\right),\\text{ }[\/latex]solve for[latex]\\text{ }x\\text{ }[\/latex]in terms of[latex]\\text{ }y.\\text{ }[\/latex]The expression for[latex]\\text{ }x\\text{ }[\/latex]is the inverse,[latex]\\text{ }{f}^{-1}\\left(y\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

    \n

    Algebraic<\/h4>\n
    \n
    \n

    6. Show that the function[latex]\\text{ }f\\left(x\\right)=a-x\\text{ }[\/latex]is its own inverse for all real numbers[latex]\\text{ }a.\\text{ }[\/latex]<\/p>\n<\/div>\n<\/div>\n

    For the following exercises, find[latex]\\text{ }{f}^{-1}\\left(x\\right)\\text{ }[\/latex]for each function.<\/p>\n

    \n
    \n

    7. [latex]f\\left(x\\right)=x+3[\/latex]<\/p>\n<\/div>\n

    \n
    Show Answer<\/span><\/p>\n
    [latex]{f}^{-1}\\left(y\\right)=y-3[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n

    8. [latex]f\\left(x\\right)=x+5[\/latex]<\/p>\n<\/div>\n<\/div>\n

    \n
    \n

    9. [latex]f\\left(x\\right)=2-x[\/latex]<\/p>\n<\/div>\n

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

    [latex]{f}^{-1}\\left(y\\right)=2-y[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

    \n
    \n

    10. [latex]f\\left(x\\right)=3-x[\/latex]<\/p>\n<\/div>\n<\/div>\n

    \n
    \n

    11. [latex]f\\left(x\\right)=\\frac{x}{x+2}[\/latex]<\/p>\n<\/div>\n

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

    [latex]{f}^{-1}\\left(y\\right)=\\frac{-2y}{y-1}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

    \n
    \n

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

    For the following exercises, find a domain on which each function[latex]\\text{ }f\\text{ }[\/latex]is one-to-one and non-decreasing. Write the domain in interval notation. Then find the inverse of[latex]\\text{ }f\\text{ }[\/latex]restricted to that domain.<\/p>\n

    \n
    \n

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

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

    domain of[latex]f\\left(x\\right):\\text{ }\\left[-7,\\infty \\right);\\text{ }{f}^{-1}\\left(y\\right)=\\sqrt{y}-7[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

    \n
    \n

    14. [latex]f\\left(x\\right)={\\left(x-6\\right)}^{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n

    \n
    \n

    15. [latex]f\\left(x\\right)={x}^{2}-5[\/latex]<\/p>\n<\/div>\n

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

    domain of[latex]\\text{ }f\\left(x\\right):\\text{ }\\left[0,\\infty \\right);\\text{ }{f}^{-1}\\left(y\\right)=\\sqrt{y+5}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

    \n
    \n

    16. Given[latex]\\text{ }f\\left(x\\right)=\\frac{x}{2+x}\\text{ }[\/latex]and[latex]\\text{ }g\\left(x\\right)=\\frac{2x}{1-x}:[\/latex]<\/p>\n

      \n
    1. Find[latex]\\text{ }f\\left(g\\left(x\\right)\\right)\\text{ }[\/latex]and[latex]\\text{ }g\\left(f\\left(x\\right)\\right).[\/latex]<\/li>\n
    2. What does the answer tell us about the relationship between[latex]\\text{ }f\\left(x\\right)\\text{ }[\/latex]and[latex]\\text{ }g\\left(x\\right)?[\/latex]<\/li>\n<\/ol>\n<\/div>\n
      \n
      Show Solution<\/span><\/p>\n
      \n

      a.[latex]\\text{ }f\\left(g\\left(x\\right)\\right)=x\\text{ }[\/latex]and[latex]\\text{ }g\\left(f\\left(x\\right)\\right)=x.\\text{ }[\/latex]b. This tells us that[latex]\\text{ }f\\text{ }[\/latex]and[latex]\\text{ }g\\text{ }[\/latex]are inverse functions<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

      For the following exercises, use function composition to verify that[latex]\\text{ }f\\left(x\\right)\\text{ }[\/latex]and[latex]\\text{ }g\\left(x\\right)\\text{ }[\/latex]are inverse functions.<\/p>\n

      \n
      \n

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

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

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

      \n
      \n

      18. [latex]f\\left(x\\right)=-3x+5\\text{ }[\/latex]and[latex]\\text{ }g\\left(x\\right)=\\frac{x-5}{-3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n

      \n

      Graphical<\/h4>\n

      For the following exercises, use a graphing utility to determine whether each function is one-to-one.<\/p>\n

      \n
      \n

      19. [latex]f\\left(x\\right)=\\sqrt{x}[\/latex]<\/p>\n<\/div>\n

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

      one-to-one<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

      \n
      \n

      20. [latex]f\\left(x\\right)=\\sqrt[3]{3x+1}[\/latex]<\/p>\n<\/div>\n<\/div>\n

      \n
      \n

      21. [latex]f\\left(x\\right)=-5x+1[\/latex]<\/p>\n<\/div>\n

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

      one-to-one<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

      \n
      \n

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

      For the following exercises, determine whether the graph represents a one-to-one function.<\/p>\n

      \n
      23.\u00a0\"Graph<\/span><\/div>\n
      \n
      Show Solution<\/span><\/p>\n
      \n

      not one-to-one<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

      \n
      24.\u00a0\"Graph<\/span><\/div>\n<\/div>\n

      For the following exercises, use the graph of[latex]\\text{ }f\\text{ }[\/latex]shown in (Figure)<\/a>.<\/p>\n

      \n
      \"Graph<\/p>\n

      Figure 11.<\/strong><\/p>\n<\/div>\n<\/div>\n

      \n
      \n

      25.Find[latex]\\text{ }f\\left(0\\right).[\/latex]<\/p>\n<\/div>\n

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

      [latex]3[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

      \n
      \n

      26. Solve[latex]\\text{ }f\\left(x\\right)=0.[\/latex]<\/p>\n<\/div>\n<\/div>\n

      \n
      \n

      27. Find[latex]\\text{ }{f}^{-1}\\left(0\\right).[\/latex]<\/p>\n<\/div>\n

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

      [latex]2[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

      \n
      \n

      28. Solve[latex]\\text{ }{f}^{-1}\\left(x\\right)=0.[\/latex]<\/p>\n<\/div>\n<\/div>\n

      For the following exercises, use the graph of the one-to-one function shown in (Figure)<\/a>.<\/p>\n

      \n
      \"Graph<\/p>\n

      Figure 12.<\/strong><\/p>\n<\/div>\n<\/div>\n

      \n
      \n

      29. Sketch the graph of[latex]\\text{ }{f}^{-1}.\\text{ }[\/latex]<\/p>\n<\/div>\n

      \n
      Show Solution<\/span><\/p>\n
      \"Graph<\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n

      30. Find[latex]\\text{ }f\\left(6\\right)\\text{ and }{f}^{-1}\\left(2\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n

      \n
      \n

      31. If the complete graph of[latex]\\text{ }f\\text{ }[\/latex]is shown, find the domain of[latex]\\text{ }f.\\text{ }[\/latex]<\/p>\n<\/div>\n

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

      [latex]\\left[2,10\\right][\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

      \n
      \n

      32. If the complete graph of[latex]\\text{ }f\\text{ }[\/latex]is shown, find the range of[latex]\\text{ }f.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n

      \n

      Numeric<\/h4>\n

      For the following exercises, evaluate or solve, assuming that the function[latex]\\text{ }f\\text{ }[\/latex]is one-to-one.<\/p>\n

      \n
      \n

      33. If[latex]\\text{ }f\\left(6\\right)=7,\\text{ }[\/latex]find[latex]\\text{ }\\text{ }{f}^{-1}\\left(7\\right).[\/latex]<\/p>\n<\/div>\n

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

      [latex]6[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

      \n
      \n

      34. If[latex]\\text{ }f\\left(3\\right)=2,\\text{ }[\/latex]find[latex]\\text{ }{f}^{-1}\\left(2\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n

      \n
      \n

      35. If[latex]\\text{ }{f}^{-1}\\left(-4\\right)=-8,\\text{ }[\/latex]find[latex]\\text{ }f\\left(-8\\right).[\/latex]<\/p>\n<\/div>\n

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

      [latex]-4[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

      \n
      \n

      36. If[latex]\\text{ }{f}^{-1}\\left(-2\\right)=-1,\\text{ }[\/latex]find[latex]\\text{ }f\\left(-1\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n

      For the following exercises, use the values listed in (Figure)<\/a> to evaluate or solve.<\/p>\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
      Table 6
      \n<\/strong>
      \n<\/strong><\/td>\n<\/tr>\n
      [latex]x[\/latex]<\/strong><\/td>\n[latex]f\\left(x\\right)[\/latex]<\/strong><\/td>\n<\/tr>\n
      0<\/td>\n8<\/td>\n<\/tr>\n
      1<\/td>\n0<\/td>\n<\/tr>\n
      2<\/td>\n7<\/td>\n<\/tr>\n
      3<\/td>\n4<\/td>\n<\/tr>\n
      4<\/td>\n2<\/td>\n<\/tr>\n
      5<\/td>\n6<\/td>\n<\/tr>\n
      6<\/td>\n5<\/td>\n<\/tr>\n
      7<\/td>\n3<\/td>\n<\/tr>\n
      8<\/td>\n9<\/td>\n<\/tr>\n
      9<\/td>\n1<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
      \n
      \n

      37. Find[latex]\\text{ }f\\left(1\\right).[\/latex]<\/p>\n<\/div>\n

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

      [latex]0[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

      \n
      \n

      38. Solve[latex]\\text{ }f\\left(x\\right)=3.[\/latex]<\/p>\n<\/div>\n<\/div>\n

      \n
      \n

      39. Find[latex]\\text{ }{f}^{-1}\\left(0\\right).[\/latex]<\/p>\n<\/div>\n

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

      [latex]\\text{ }1\\text{ }[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

      \n
      \n

      40. Solve[latex]\\text{ }{f}^{-1}\\left(x\\right)=7.[\/latex]<\/p>\n<\/div>\n<\/div>\n

      \n
      \n

      41. Use the tabular representation of[latex]\\text{ }f\\text{ }[\/latex]in (Figure)<\/a> to create a table for[latex]\\text{ }{f}^{-1}\\left(x\\right).[\/latex]<\/p>\n\n\n\n\n\n
      Table 7<\/strong><\/td>\n<\/tr>\n
      [latex]x[\/latex]<\/strong><\/td>\n3<\/td>\n6<\/td>\n9<\/td>\n13<\/td>\n14<\/td>\n<\/tr>\n
      [latex]f\\left(x\\right)[\/latex]<\/strong><\/td>\n1<\/td>\n4<\/td>\n7<\/td>\n12<\/td>\n16<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n
      \n
      Show Solution<\/span><\/p>\n
      \n\n\n\n\n
      [latex]x[\/latex]<\/td>\n1<\/td>\n4<\/td>\n7<\/td>\n12<\/td>\n16<\/td>\n<\/tr>\n
      [latex]{f}^{-1}\\left(x\\right)[\/latex]<\/td>\n3<\/td>\n6<\/td>\n9<\/td>\n13<\/td>\n14<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n

      Technology<\/h4>\n

      For the following exercises, find the inverse function. Then, graph the function and its inverse.<\/p>\n

      \n
      \n

      42. [latex]f\\left(x\\right)=\\frac{3}{x-2}[\/latex]<\/p>\n<\/div>\n<\/div>\n

      \n
      \n

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

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

      [latex]{f}^{-1}\\left(x\\right)={\\left(1+x\\right)}^{1\/3}[\/latex]<\/p>\n

      \"Graph<\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n

      \n
      \n

      44. Find the inverse function of[latex]\\text{ }f\\left(x\\right)=\\frac{1}{x-1}.\\text{ }[\/latex]Use a graphing utility to find its domain and range. Write the domain and range in interval notation.<\/p>\n<\/div>\n<\/div>\n<\/div>\n

      \n

      Real-World Applications<\/h4>\n
      \n
      \n

      45. To convert from[latex]\\text{ }x\\text{ }[\/latex]degrees Celsius to[latex]\\text{ }y\\text{ }[\/latex]degrees Fahrenheit, we use the formula[latex]\\text{ }f\\left(x\\right)=\\frac{9}{5}x+32.\\text{ }[\/latex]Find the inverse function, if it exists, and explain its meaning.<\/p>\n<\/div>\n

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

      [latex]{f}^{-1}\\left(x\\right)=\\frac{5}{9}\\left(x-32\\right).\\text{ }[\/latex]Given the Fahrenheit temperature,[latex]\\text{ }x,\\text{ }[\/latex]this formula allows you to calculate the Celsius temperature.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

      \n
      \n

      46. The circumference[latex]\\text{ }C\\text{ }[\/latex]of a circle is a function of its radius given by[latex]\\text{ }C\\left(r\\right)=2\\pi r.\\text{ }[\/latex]Express the radius of a circle as a function of its circumference. Call this function[latex]\\text{ }r\\left(C\\right).\\text{ }[\/latex]Find[latex]\\text{ }r\\left(36\\pi \\right)\\text{ }[\/latex]and interpret its meaning.<\/p>\n<\/div>\n<\/div>\n

      \n
      \n

      47. A car travels at a constant speed of 50 miles per hour. The distance the car travels in miles is a function of time,[latex]\\text{ }t,\\text{ }[\/latex]in hours given by[latex]\\text{ }d\\left(t\\right)=50t.\\text{ }[\/latex]Find the inverse function by expressing the time of travel in terms of the distance traveled. Call this function[latex]\\text{ }t\\left(d\\right).\\text{ }[\/latex]Find[latex]\\text{ }t\\left(180\\right)\\text{ }[\/latex]and interpret its meaning.<\/p>\n<\/div>\n

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

      [latex]t\\left(d\\right)=\\frac{d}{50},\\text{ }[\/latex][latex]t\\left(180\\right)=\\frac{180}{50}.\\text{ }[\/latex]The time for the car to travel 180 miles is 3.6 hours.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"author":158108,"menu_order":2,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2355","chapter","type-chapter","status-web-only","hentry"],"part":223,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/2355","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/users\/158108"}],"version-history":[{"count":2,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/2355\/revisions"}],"predecessor-version":[{"id":2357,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/2355\/revisions\/2357"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/parts\/223"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/2355\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/media?parent=2355"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=2355"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/contributor?post=2355"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/license?post=2355"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}