{"id":2287,"date":"2019-05-09T15:41:31","date_gmt":"2019-05-09T15:41:31","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/?post_type=chapter&p=2287"},"modified":"2020-01-14T17:29:43","modified_gmt":"2020-01-14T17:29:43","slug":"1-2-section-exercises","status":"web-only","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/chapter\/1-2-section-exercises\/","title":{"raw":"1.2 Section Exercises","rendered":"1.2 Section Exercises"},"content":{"raw":"
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1.2 Section Exercises<\/h3>\r\n
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Verbal<\/h4>\r\n
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1. Why does the domain differ for different functions?<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165134199600\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134199600\"]\r\n

The domain of a function depends upon what values of the independent variable make the function undefined or imaginary.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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2. How do we determine the domain of a function defined by an equation?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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3. Explain why the domain of[latex]\\text{ }f\\left(x\\right)=\\sqrt[3]{x}\\text{ }[\/latex]is different from the domain of[latex]\\text{ }f\\left(x\\right)=\\sqrt[]{x}.[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137727146\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137727146\"]\r\n

There is no restriction on[latex]\\text{ }x\\text{ }[\/latex]for[latex]\\text{ }f\\left(x\\right)=\\sqrt[3]{x}\\text{ }[\/latex]because you can take the cube root of any real number. So the domain is all real numbers,[latex]\\text{ }\\left(-\\infty ,\\infty \\right).\\text{ }[\/latex]When dealing with the set of real numbers, you cannot take the square root of negative numbers. So[latex]\\text{ }x[\/latex]-values are restricted for[latex]\\text{ }f\\left(x\\right)=\\sqrt[]{x}\\text{ }[\/latex]to nonnegative numbers and the domain is[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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4. When describing sets of numbers using interval notation, when do you use a parenthesis and when do you use a bracket?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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5. How do you graph a piecewise function?<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137574335\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137574335\"]\r\n

Graph each formula of the piecewise function over its corresponding domain. Use the same scale for the[latex]\\text{ }x[\/latex]-axis and[latex]\\text{ }y[\/latex]-axis for each graph. Indicate inclusive endpoints with a solid circle and exclusive endpoints with an open circle. Use an arrow to indicate[latex]\\text{ }-\\infty \\text{ }[\/latex]or[latex]\\text{ }\\text{ }\\infty .\\text{ }[\/latex]Combine the graphs to find the graph of the piecewise function.<\/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

For the following exercises, find the domain of each function using interval notation.<\/p>\r\n\r\n

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6. [latex]f\\left(x\\right)=-2x\\left(x-1\\right)\\left(x-2\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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

[reveal-answer q=\"fs-id1165137731586\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137731586\"]\r\n

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

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

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9. [latex]f\\left(x\\right)=3-\\sqrt{6-2x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137451053\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137451053\"]\r\n

[latex]\\left(-\\infty ,3\\right][\/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)=\\sqrt{4-3x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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11. [latex]\\begin{array}{l}\\\\ f\\left(x\\right)=\\sqrt[]{{x}^{2}+4}\\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165134192936\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134192936\"]\r\n

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

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

[reveal-answer q=\"fs-id1165134058389\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134058389\"]\r\n

[latex]\\left(-\\infty ,\\infty \\right)[\/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)=\\frac{9}{x-6}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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

[reveal-answer q=\"fs-id1165137454548\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137454548\"]\r\n

[latex]\\left(-\\infty ,-\\frac{1}{2}\\right)\\cup \\left(-\\frac{1}{2},\\infty \\right)[\/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{\\sqrt{x+4}}{x-4}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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17. [latex]f\\left(x\\right)=\\frac{x-3}{{x}^{2}+9x-22}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137528909\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137528909\"]\r\n

[latex]\\left(-\\infty ,-11\\right)\\cup \\left(-11,2\\right)\\cup \\left(2,\\infty \\right)[\/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)=\\frac{1}{{x}^{2}-x-6}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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19. [latex]f\\left(x\\right)=\\frac{2{x}^{3}-250}{{x}^{2}-2x-15}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135256053\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135256053\"]\r\n

[latex]\\left(-\\infty ,-3\\right)\\cup \\left(-3,5\\right)\\cup \\left(5,\\infty \\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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20. [latex]\\frac{5}{\\sqrt{x-3}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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21. [latex]\\frac{2x+1}{\\sqrt{5-x}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137647829\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137647829\"]\r\n

[latex]\\left(-\\infty ,5\\right)[\/latex]<\/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)=\\frac{\\sqrt{x-4}}{\\sqrt{x-6}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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23. [latex]f\\left(x\\right)=\\frac{\\sqrt{x-6}}{\\sqrt{x-4}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137611238\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137611238\"]\r\n

[latex]\\left[6,\\infty \\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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

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25. [latex]f\\left(x\\right)=\\frac{{x}^{2}-9x}{{x}^{2}-81}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135188135\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135188135\"]\r\n

[latex]\\left(-\\infty ,-9\\right)\\cup \\left(-9,9\\right)\\cup \\left(9,\\infty \\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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26. Find the domain of the function[latex]\\text{ }f\\left(x\\right)=\\sqrt{2{x}^{3}-50x}\\text{ }[\/latex]by:<\/p>\r\n\r\n

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  1. using algebra.<\/li>\r\n \t
  2. graphing the function in the radicand and determining intervals on the x<\/em>-axis for which the radicand is nonnegative.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n
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    Graphical<\/h4>\r\n

    For the following exercises, write the domain and range of each function using interval notation.<\/p>\r\n\r\n

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    27.<\/span><\/div>\r\n
    \u00a0\"Graph<\/span><\/div>\r\n
    [reveal-answer q=\"fs-id1165137820038\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137820038\"]\r\n

    domain:[latex]\\text{ }\\left(2,8\\right],\\text{ }[\/latex]range[latex]\\text{ }\\left[6,8\\right)\\text{ }[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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    28.\u00a0<\/span><\/div>\r\n
    \"Graph<\/div>\r\n<\/div>\r\n
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    29.\u00a0<\/span><\/div>\r\n
    \"Graph<\/div>\r\n
    [reveal-answer q=\"fs-id1165137847285\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137847285\"]\r\n

    domain:[latex]\\text{ }\\left[-4,\\text{ 4],}\\text{ }[\/latex]range:[latex]\\text{ }\\left[0,\\text{ 2]}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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    30.\u00a0<\/span><\/div>\r\n
    \"Graph<\/div>\r\n<\/div>\r\n
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    31.\u00a0<\/span><\/div>\r\n
    \"Graph<\/div>\r\n
    [reveal-answer q=\"fs-id1165137657479\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137657479\"]\r\n

    domain:[latex]\\text{ }\\left[-5,\\text{ }3\\right),\\text{ }[\/latex]range:[latex]\\text{ }\\left[0,2\\right][\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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    32.\u00a0<\/span><\/div>\r\n
    \"Graph<\/div>\r\n<\/div>\r\n
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    33.\u00a0<\/span><\/div>\r\n
    \"Graph<\/div>\r\n
    [reveal-answer q=\"fs-id1165137445711\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137445711\"]\r\n

    domain:[latex]\\text{ }\\left(-\\infty ,1\\right],\\text{ }[\/latex]range:[latex]\\text{ }\\left[0,\\infty \\right)\\text{ }[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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    34.\u00a0<\/span><\/div>\r\n
    \"Graph<\/div>\r\n<\/div>\r\n
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    35.<\/span><\/div>\r\n
    \"Graph<\/span><\/div>\r\n
    [reveal-answer q=\"fs-id1165134043582\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134043582\"]\r\n

    domain:[latex]\\text{ }\\left[-6,-\\frac{1}{6}\\right]\\cup \\left[\\frac{1}{6},6\\right];\\text{ }[\/latex]range:[latex]\\text{ }\\left[-6,-\\frac{1}{6}\\right]\\cup \\left[\\frac{1}{6},6\\right]\\text{ }[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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    36.\u00a0<\/span><\/div>\r\n
    \"Graph<\/div>\r\n<\/div>\r\n
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    37.\u00a0<\/span><\/div>\r\n
    \"Graph<\/div>\r\n
    [reveal-answer q=\"fs-id1165137575572\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137575572\"]\r\n

    domain:[latex]\\text{ }\\left[-3,\\text{ }\\infty \\right);\\text{ }[\/latex]range:[latex]\\text{ }\\left[0,\\infty \\right)\\text{ }[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

    For the following exercises, sketch a graph of the piecewise function. Write the domain in interval notation.<\/p>\r\n\r\n

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    38. [latex]f\\left(x\\right)=\\bigg\\{\\begin{array}{lll}x+1\\hfill & \\text{if}\\hfill & x<-2\\hfill \\\\ -2x-3\\hfill & \\text{if}\\hfill & x\\ge -2\\hfill \\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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    39. [latex]f\\left(x\\right)=\\bigg\\{\\begin{array}{lll}2x-1\\hfill & \\text{if}\\hfill & x<1\\hfill \\\\ 1+x\\hfill & \\text{if}\\hfill & x\\ge 1\\hfill \\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

    [reveal-answer q=\"fs-id1165135481131\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135481131\"]\r\n

    domain:[latex]\\text{ }\\left(-\\infty ,\\infty \\right)[\/latex]<\/p>\r\n\"graph<\/span>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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    40. [latex]f\\left(x\\right)=\\bigg\\{\\begin{array}{c}x+1\\text{ }\\text{ }\\text{if}\\text{ }\\text{ }x<0\\\\ x-1\\text{ }\\text{ }\\text{if}\\text{ }\\text{ }\\text{ }x>0\\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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    41. [latex]f\\left(x\\right)=\\bigg\\{\\begin{array}{ccc}3& \\text{if}& x<0\\\\ \\sqrt{x}& \\text{if}& x\\ge 0\\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

    [reveal-answer q=\"fs-id1165137500956\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137500956\"]\r\n

    domain:[latex]\\text{ }\\left(-\\infty ,\\infty \\right)[\/latex]<\/p>\r\n\"graph<\/span>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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    42. [latex]f\\left(x\\right)=\\bigg\\{\\begin{array}{c}{x}^{2}\\text{ if }x<0\\\\ 1-x\\text{ if }x>0\\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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    43. [latex]f\\left(x\\right)=\\bigg\\{\\begin{array}{r}\\hfill \\begin{array}{r}\\hfill {x}^{2}\\\\ \\hfill x+2\\end{array}\\end{array}\\text{ }\\text{ }\\begin{array}{l}\\text{if}\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }x<0\\hfill \\\\ \\text{if}\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }x\\ge 0\\hfill \\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

    [reveal-answer q=\"fs-id1165137667233\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137667233\"]\r\n

    domain:[latex]\\text{ }\\left(-\\infty ,\\infty \\right)[\/latex]<\/p>\r\n\"graph<\/span>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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    44. [latex]f\\left(x\\right)=\\bigg\\{\\begin{array}{ccc}x+1& \\text{if}& x<1\\\\ {x}^{3}& \\text{if}& x\\ge 1\\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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    45. [latex]f\\left(x\\right)=\\bigg\\{\\begin{array}{c}|x|\\\\ 1\\end{array}\\begin{array}{l}\\text{ }\\text{ }\\text{ }\\text{if}\\text{ }\\text{ }\\text{ }x<2\\hfill \\\\ \\text{ }\\text{ }\\text{ }\\text{if}\\text{ }\\text{ }\\text{ }x\\ge 2\\hfill \\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

    [reveal-answer q=\"fs-id1165137401041\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137401041\"]\r\n

    domain:[latex]\\text{ }\\left(-\\infty ,\\infty \\right)[\/latex]<\/p>\r\n\"\"<\/span>[\/hidden-answer]\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, given each function [latex]f,[\/latex]evaluate [latex]f\\left(-3\\right),\\text{ }f\\left(-2\\right),\\text{ }f\\left(-1\\right),[\/latex] and [latex]f\\left(0\\right).[\/latex]<\/p>\r\n\r\n

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    46. [latex]f\\left(x\\right)=\\bigg\\{\\begin{array}{lll}x+1\\hfill & \\text{if}\\hfill & x<-2\\hfill \\\\ -2x-3\\hfill & \\text{if}\\hfill & x\\ge -2\\hfill \\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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    47. [latex]f\\left(x\\right)=\\bigg\\{\\begin{array}{cc}1& \\text{if }x\\le -3\\\\ 0& \\text{if }x>-3\\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

    [reveal-answer q=\"fs-id1165137804494\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137804494\"]\r\n

    [latex]\\begin{array}{cccc}f\\left(-3\\right)=1;& f\\left(-2\\right)=0;& f\\left(-1\\right)=0;& f\\left(0\\right)=0\\end{array}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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    48. [latex]f\\left(x\\right)=\\bigg\\{\\begin{array}{cc}-2{x}^{2}+3& \\text{if }x\\le -1\\\\ 5x-7& \\text{if }x>-1\\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

    For the following exercises, given each function[latex]\\text{ }f,\\text{ }[\/latex]evaluate[latex]f\\left(-1\\right),\\text{ }f\\left(0\\right),\\text{ }f\\left(2\\right),\\text{ }[\/latex]and[latex]\\text{ }f\\left(4\\right).[\/latex]<\/p>\r\n\r\n

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    49. [latex]f\\left(x\\right)=\\bigg\\{\\begin{array}{lll}7x+3\\hfill & \\text{if}\\hfill & x<0\\hfill \\\\ 7x+6\\hfill & \\text{if}\\hfill & x\\ge 0\\hfill \\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

    [reveal-answer q=\"fs-id1165137476514\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137476514\"]\r\n

    [latex]\\begin{array}{cccc}f\\left(-1\\right)=-4;& f\\left(0\\right)=6;& f\\left(2\\right)=20;& f\\left(4\\right)=34\\end{array}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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    50. [latex]f\\left(x\\right)=\\bigg\\{\\begin{array}{ccc}{x}^{2}-2& \\text{if}& x<2\\\\ 4+|x-5|& \\text{if}& x\\ge 2\\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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    51. [latex]f\\left(x\\right)=\\Bigg\\{\\begin{array}{ccc}5x& \\text{if}& x<0\\\\ 3& \\text{if}& 0\\le x\\le 3\\\\ {x}^{2}& \\text{if}& x>3\\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

    [reveal-answer q=\"fs-id1165135699157\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135699157\"]\r\n

    [latex]\\begin{array}{cccc}f\\left(-1\\right)=-5;& f\\left(0\\right)=3;& f\\left(2\\right)=3;& f\\left(4\\right)=16\\end{array}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

    For the following exercises, write the domain for the piecewise function in interval notation.<\/p>\r\n\r\n

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    52. [latex]f\\left(x\\right)=\\bigg\\{\\begin{array}{c}x+1\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ if}\\text{ }\\text{ }x<-2\\\\ -2x-3\\text{ }\\text{ }\\text{if}\\text{ }\\text{ }x\\ge -2\\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

    \r\n
    \r\n

    53. [latex]f\\left(x\\right)=\\bigg\\{\\begin{array}{c}{x}^{2}-2\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ if}\\text{ }\\text{ }x<1\\\\ -{x}^{2}+2\\text{ }\\text{ }\\text{if}\\text{ }\\text{ }x>1\\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

    [reveal-answer q=\"fs-id1165135420410\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135420410\"]\r\n

    domain:[latex]\\text{ }\\left(-\\infty ,1\\right)\\cup \\left(1,\\infty \\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

    \r\n
    \r\n

    54. [latex]f\\left(x\\right)=\\bigg\\{\\begin{array}{c}2x-3\\\\ -3{x}^{2}\\end{array}\\text{ }\\text{ }\\begin{array}{c}\\text{if}\\text{ }\\text{ }\\text{ }x<0\\\\ \\text{if}\\text{ }\\text{ }\\text{ }x\\ge 2\\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

    \r\n

    Technology<\/h4>\r\n
    \r\n
    \r\n

    55. Graph[latex]\\text{ }y=\\frac{1}{{x}^{2}}\\text{ }[\/latex]on the viewing window[latex]\\text{ }\\left[-0.5,-0.1\\right]\\text{ }[\/latex]and[latex]\\text{ }\\left[0.1,0.5\\right].\\text{ }[\/latex]Determine the corresponding range for the viewing window. Show the graphs.<\/p>\r\n\r\n<\/div>\r\n

    [reveal-answer q=\"fs-id1165137501974\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137501974\"]<\/div>\r\n
    \r\n

    \"Graph<\/p>\r\nwindow:[latex]\\text{ }\\left[-0.5,-0.1\\right];\\text{ }[\/latex]range:[latex]\\text{ }\\left[4,\\text{ }100\\right][\/latex]\r\n\r\n\"Graph\r\n\r\n<\/div>\r\n<\/div>\r\n \r\n

    \r\n
    \r\n

    window:[latex]\\text{ }\\left[0.1,\\text{ }0.5\\right];\\text{ }[\/latex]range:[latex]\\text{ }\\left[4,\\text{ }100\\right][\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

    \r\n
    \r\n

    56. Graph[latex]\\text{ }y=\\frac{1}{x}\\text{ }[\/latex]on the viewing window[latex]\\text{ }\\left[-0.5,-0.1\\right]\\text{ }[\/latex]and[latex]\\text{ }\\left[0.1,\\text{ }0.5\\right].\\text{ }[\/latex]Determine the corresponding range for the viewing window. Show the graphs.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

    \r\n

    Extension<\/h4>\r\n
    \r\n
    \r\n

    57. Suppose the range of a function[latex]\\text{ }f\\text{ }[\/latex]is[latex]\\text{ }\\left[-5,\\text{ }8\\right].\\text{ }[\/latex]What is the range of[latex]\\text{ }|f\\left(x\\right)|?[\/latex]<\/p>\r\n\r\n<\/div>\r\n

    [reveal-answer q=\"fs-id1165134555582\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134555582\"]\r\n

    [latex]\\left[0,\\text{ }8\\right][\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

    \r\n
    \r\n

    58. Create a function in which the range is all nonnegative real numbers.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

    \r\n
    \r\n

    59. Create a function in which the domain is[latex]\\text{ }x>2.[\/latex]<\/p>\r\n\r\n<\/div>\r\n

    [reveal-answer q=\"fs-id1165133210812\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165133210812\"]\r\n

    Many answers. One function is[latex]\\text{ }f\\left(x\\right)=\\frac{1}{\\sqrt{x-2}}.[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

    \r\n

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

    60. The height[latex]\\text{ }h\\text{ }[\/latex]of a projectile is a function of the time[latex]\\text{ }t\\text{ }[\/latex]it is in the air. The height in feet for[latex]\\text{ }t\\text{ }[\/latex]seconds is given by the function [latex]h\\left(t\\right)=-16{t}^{2}+96t.[\/latex] What is the domain of the function? What does the domain mean in the context of the problem?<\/p>\r\n\r\n<\/div>\r\n

    [reveal-answer q=\"fs-id1165137446701\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137446701\"]\r\n

    The domain is[latex]\\text{ }\\left[0,\\text{ }6\\right];\\text{ }[\/latex]it takes 6 seconds for the projectile to leave the ground and return to the ground<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

    \r\n
    \r\n

    61. The cost in dollars of making[latex]\\text{ }x\\text{ }[\/latex]items is given by the function[latex]\\text{ }C\\left(x\\right)=10x+500.[\/latex]<\/p>\r\n\r\n

      \r\n \t
    1. The fixed cost is determined when zero items are produced. Find the fixed cost for this item.<\/li>\r\n \t
    2. What is the cost of making 25 items?<\/li>\r\n \t
    3. Suppose the maximum cost allowed is $1500. What are the domain and range of the cost function,[latex]\\text{ }C\\left(x\\right)?[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n ","rendered":"
      \n

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

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

      1. Why does the domain differ for different functions?<\/p>\n<\/div>\n

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

      The domain of a function depends upon what values of the independent variable make the function undefined or imaginary.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

      \n
      \n

      2. How do we determine the domain of a function defined by an equation?<\/p>\n<\/div>\n<\/div>\n

      \n
      \n

      3. Explain why the domain of[latex]\\text{ }f\\left(x\\right)=\\sqrt[3]{x}\\text{ }[\/latex]is different from the domain of[latex]\\text{ }f\\left(x\\right)=\\sqrt[]{x}.[\/latex]<\/p>\n<\/div>\n

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

      There is no restriction on[latex]\\text{ }x\\text{ }[\/latex]for[latex]\\text{ }f\\left(x\\right)=\\sqrt[3]{x}\\text{ }[\/latex]because you can take the cube root of any real number. So the domain is all real numbers,[latex]\\text{ }\\left(-\\infty ,\\infty \\right).\\text{ }[\/latex]When dealing with the set of real numbers, you cannot take the square root of negative numbers. So[latex]\\text{ }x[\/latex]-values are restricted for[latex]\\text{ }f\\left(x\\right)=\\sqrt[]{x}\\text{ }[\/latex]to nonnegative numbers and the domain is[latex]\\text{ }\\left[0,\\infty \\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

      \n
      \n

      4. When describing sets of numbers using interval notation, when do you use a parenthesis and when do you use a bracket?<\/p>\n<\/div>\n<\/div>\n

      \n
      \n

      5. How do you graph a piecewise function?<\/p>\n<\/div>\n

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

      Graph each formula of the piecewise function over its corresponding domain. Use the same scale for the[latex]\\text{ }x[\/latex]-axis and[latex]\\text{ }y[\/latex]-axis for each graph. Indicate inclusive endpoints with a solid circle and exclusive endpoints with an open circle. Use an arrow to indicate[latex]\\text{ }-\\infty \\text{ }[\/latex]or[latex]\\text{ }\\text{ }\\infty .\\text{ }[\/latex]Combine the graphs to find the graph of the piecewise function.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

      \n

      Algebraic<\/h4>\n

      For the following exercises, find the domain of each function using interval notation.<\/p>\n

      \n
      \n

      6. [latex]f\\left(x\\right)=-2x\\left(x-1\\right)\\left(x-2\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

      \n
      \n

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

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

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

      \n
      \n

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

      \n
      \n

      9. [latex]f\\left(x\\right)=3-\\sqrt{6-2x}[\/latex]<\/p>\n<\/div>\n

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

      [latex]\\left(-\\infty ,3\\right][\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

      \n
      \n

      10. [latex]f\\left(x\\right)=\\sqrt{4-3x}[\/latex]<\/p>\n<\/div>\n<\/div>\n

      \n
      \n

      11. [latex]\\begin{array}{l}\\\\ f\\left(x\\right)=\\sqrt[]{{x}^{2}+4}\\end{array}[\/latex]<\/p>\n<\/div>\n

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

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

      \n
      \n

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

      \n
      \n

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

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

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

      \n
      \n

      14. [latex]f\\left(x\\right)=\\frac{9}{x-6}[\/latex]<\/p>\n<\/div>\n<\/div>\n

      \n
      \n

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

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

      [latex]\\left(-\\infty ,-\\frac{1}{2}\\right)\\cup \\left(-\\frac{1}{2},\\infty \\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

      \n
      \n

      16. [latex]f\\left(x\\right)=\\frac{\\sqrt{x+4}}{x-4}[\/latex]<\/p>\n<\/div>\n<\/div>\n

      \n
      \n

      17. [latex]f\\left(x\\right)=\\frac{x-3}{{x}^{2}+9x-22}[\/latex]<\/p>\n<\/div>\n

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

      [latex]\\left(-\\infty ,-11\\right)\\cup \\left(-11,2\\right)\\cup \\left(2,\\infty \\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

      \n
      \n

      18. [latex]f\\left(x\\right)=\\frac{1}{{x}^{2}-x-6}[\/latex]<\/p>\n<\/div>\n<\/div>\n

      \n
      \n

      19. [latex]f\\left(x\\right)=\\frac{2{x}^{3}-250}{{x}^{2}-2x-15}[\/latex]<\/p>\n<\/div>\n

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

      [latex]\\left(-\\infty ,-3\\right)\\cup \\left(-3,5\\right)\\cup \\left(5,\\infty \\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

      \n
      \n

      20. [latex]\\frac{5}{\\sqrt{x-3}}[\/latex]<\/p>\n<\/div>\n<\/div>\n

      \n
      \n

      21. [latex]\\frac{2x+1}{\\sqrt{5-x}}[\/latex]<\/p>\n<\/div>\n

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

      [latex]\\left(-\\infty ,5\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

      \n
      \n

      22. [latex]f\\left(x\\right)=\\frac{\\sqrt{x-4}}{\\sqrt{x-6}}[\/latex]<\/p>\n<\/div>\n<\/div>\n

      \n
      \n

      23. [latex]f\\left(x\\right)=\\frac{\\sqrt{x-6}}{\\sqrt{x-4}}[\/latex]<\/p>\n<\/div>\n

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

      [latex]\\left[6,\\infty \\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

      \n
      \n

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

      \n
      \n

      25. [latex]f\\left(x\\right)=\\frac{{x}^{2}-9x}{{x}^{2}-81}[\/latex]<\/p>\n<\/div>\n

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

      [latex]\\left(-\\infty ,-9\\right)\\cup \\left(-9,9\\right)\\cup \\left(9,\\infty \\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

      \n
      \n

      26. Find the domain of the function[latex]\\text{ }f\\left(x\\right)=\\sqrt{2{x}^{3}-50x}\\text{ }[\/latex]by:<\/p>\n

        \n
      1. using algebra.<\/li>\n
      2. graphing the function in the radicand and determining intervals on the x<\/em>-axis for which the radicand is nonnegative.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n
        \n

        Graphical<\/h4>\n

        For the following exercises, write the domain and range of each function using interval notation.<\/p>\n

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

        domain:[latex]\\text{ }\\left(2,8\\right],\\text{ }[\/latex]range[latex]\\text{ }\\left[6,8\\right)\\text{ }[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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

        domain:[latex]\\text{ }\\left[-4,\\text{ 4],}\\text{ }[\/latex]range:[latex]\\text{ }\\left[0,\\text{ 2]}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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

        domain:[latex]\\text{ }\\left[-5,\\text{ }3\\right),\\text{ }[\/latex]range:[latex]\\text{ }\\left[0,2\\right][\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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

        domain:[latex]\\text{ }\\left(-\\infty ,1\\right],\\text{ }[\/latex]range:[latex]\\text{ }\\left[0,\\infty \\right)\\text{ }[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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

        domain:[latex]\\text{ }\\left[-6,-\\frac{1}{6}\\right]\\cup \\left[\\frac{1}{6},6\\right];\\text{ }[\/latex]range:[latex]\\text{ }\\left[-6,-\\frac{1}{6}\\right]\\cup \\left[\\frac{1}{6},6\\right]\\text{ }[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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

        domain:[latex]\\text{ }\\left[-3,\\text{ }\\infty \\right);\\text{ }[\/latex]range:[latex]\\text{ }\\left[0,\\infty \\right)\\text{ }[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

        For the following exercises, sketch a graph of the piecewise function. Write the domain in interval notation.<\/p>\n

        \n
        \n

        38. [latex]f\\left(x\\right)=\\bigg\\{\\begin{array}{lll}x+1\\hfill & \\text{if}\\hfill & x<-2\\hfill \\\\ -2x-3\\hfill & \\text{if}\\hfill & x\\ge -2\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n

        \n
        \n

        39. [latex]f\\left(x\\right)=\\bigg\\{\\begin{array}{lll}2x-1\\hfill & \\text{if}\\hfill & x<1\\hfill \\\\ 1+x\\hfill & \\text{if}\\hfill & x\\ge 1\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n

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

        domain:[latex]\\text{ }\\left(-\\infty ,\\infty \\right)[\/latex]<\/p>\n

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

        \n
        \n

        40. [latex]f\\left(x\\right)=\\bigg\\{\\begin{array}{c}x+1\\text{ }\\text{ }\\text{if}\\text{ }\\text{ }x<0\\\\ x-1\\text{ }\\text{ }\\text{if}\\text{ }\\text{ }\\text{ }x>0\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n

        \n
        \n

        41. [latex]f\\left(x\\right)=\\bigg\\{\\begin{array}{ccc}3& \\text{if}& x<0\\\\ \\sqrt{x}& \\text{if}& x\\ge 0\\end{array}[\/latex]<\/p>\n<\/div>\n

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

        domain:[latex]\\text{ }\\left(-\\infty ,\\infty \\right)[\/latex]<\/p>\n

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

        \n
        \n

        42. [latex]f\\left(x\\right)=\\bigg\\{\\begin{array}{c}{x}^{2}\\text{ if }x<0\\\\ 1-x\\text{ if }x>0\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n

        \n
        \n

        43. [latex]f\\left(x\\right)=\\bigg\\{\\begin{array}{r}\\hfill \\begin{array}{r}\\hfill {x}^{2}\\\\ \\hfill x+2\\end{array}\\end{array}\\text{ }\\text{ }\\begin{array}{l}\\text{if}\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }x<0\\hfill \\\\ \\text{if}\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }x\\ge 0\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n

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

        domain:[latex]\\text{ }\\left(-\\infty ,\\infty \\right)[\/latex]<\/p>\n

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

        \n
        \n

        44. [latex]f\\left(x\\right)=\\bigg\\{\\begin{array}{ccc}x+1& \\text{if}& x<1\\\\ {x}^{3}& \\text{if}& x\\ge 1\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n

        \n
        \n

        45. [latex]f\\left(x\\right)=\\bigg\\{\\begin{array}{c}|x|\\\\ 1\\end{array}\\begin{array}{l}\\text{ }\\text{ }\\text{ }\\text{if}\\text{ }\\text{ }\\text{ }x<2\\hfill \\\\ \\text{ }\\text{ }\\text{ }\\text{if}\\text{ }\\text{ }\\text{ }x\\ge 2\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n

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

        domain:[latex]\\text{ }\\left(-\\infty ,\\infty \\right)[\/latex]<\/p>\n

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

        \n

        Numeric<\/h4>\n

        For the following exercises, given each function [latex]f,[\/latex]evaluate [latex]f\\left(-3\\right),\\text{ }f\\left(-2\\right),\\text{ }f\\left(-1\\right),[\/latex] and [latex]f\\left(0\\right).[\/latex]<\/p>\n

        \n
        \n

        46. [latex]f\\left(x\\right)=\\bigg\\{\\begin{array}{lll}x+1\\hfill & \\text{if}\\hfill & x<-2\\hfill \\\\ -2x-3\\hfill & \\text{if}\\hfill & x\\ge -2\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n

        \n
        \n

        47. [latex]f\\left(x\\right)=\\bigg\\{\\begin{array}{cc}1& \\text{if }x\\le -3\\\\ 0& \\text{if }x>-3\\end{array}[\/latex]<\/p>\n<\/div>\n

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

        [latex]\\begin{array}{cccc}f\\left(-3\\right)=1;& f\\left(-2\\right)=0;& f\\left(-1\\right)=0;& f\\left(0\\right)=0\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

        \n
        \n

        48. [latex]f\\left(x\\right)=\\bigg\\{\\begin{array}{cc}-2{x}^{2}+3& \\text{if }x\\le -1\\\\ 5x-7& \\text{if }x>-1\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n

        For the following exercises, given each function[latex]\\text{ }f,\\text{ }[\/latex]evaluate[latex]f\\left(-1\\right),\\text{ }f\\left(0\\right),\\text{ }f\\left(2\\right),\\text{ }[\/latex]and[latex]\\text{ }f\\left(4\\right).[\/latex]<\/p>\n

        \n
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        49. [latex]f\\left(x\\right)=\\bigg\\{\\begin{array}{lll}7x+3\\hfill & \\text{if}\\hfill & x<0\\hfill \\\\ 7x+6\\hfill & \\text{if}\\hfill & x\\ge 0\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n

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

        [latex]\\begin{array}{cccc}f\\left(-1\\right)=-4;& f\\left(0\\right)=6;& f\\left(2\\right)=20;& f\\left(4\\right)=34\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

        \n
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        50. [latex]f\\left(x\\right)=\\bigg\\{\\begin{array}{ccc}{x}^{2}-2& \\text{if}& x<2\\\\ 4+|x-5|& \\text{if}& x\\ge 2\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n

        \n
        \n

        51. [latex]f\\left(x\\right)=\\Bigg\\{\\begin{array}{ccc}5x& \\text{if}& x<0\\\\ 3& \\text{if}& 0\\le x\\le 3\\\\ {x}^{2}& \\text{if}& x>3\\end{array}[\/latex]<\/p>\n<\/div>\n

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

        [latex]\\begin{array}{cccc}f\\left(-1\\right)=-5;& f\\left(0\\right)=3;& f\\left(2\\right)=3;& f\\left(4\\right)=16\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

        For the following exercises, write the domain for the piecewise function in interval notation.<\/p>\n

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        \n

        52. [latex]f\\left(x\\right)=\\bigg\\{\\begin{array}{c}x+1\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ if}\\text{ }\\text{ }x<-2\\\\ -2x-3\\text{ }\\text{ }\\text{if}\\text{ }\\text{ }x\\ge -2\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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        53. [latex]f\\left(x\\right)=\\bigg\\{\\begin{array}{c}{x}^{2}-2\\text{ }\\text{ }\\text{ }\\text{ }\\text{ }\\text{ if}\\text{ }\\text{ }x<1\\\\ -{x}^{2}+2\\text{ }\\text{ }\\text{if}\\text{ }\\text{ }x>1\\end{array}[\/latex]<\/p>\n<\/div>\n

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

        domain:[latex]\\text{ }\\left(-\\infty ,1\\right)\\cup \\left(1,\\infty \\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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        \n

        54. [latex]f\\left(x\\right)=\\bigg\\{\\begin{array}{c}2x-3\\\\ -3{x}^{2}\\end{array}\\text{ }\\text{ }\\begin{array}{c}\\text{if}\\text{ }\\text{ }\\text{ }x<0\\\\ \\text{if}\\text{ }\\text{ }\\text{ }x\\ge 2\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n

        \n

        Technology<\/h4>\n
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        \n

        55. Graph[latex]\\text{ }y=\\frac{1}{{x}^{2}}\\text{ }[\/latex]on the viewing window[latex]\\text{ }\\left[-0.5,-0.1\\right]\\text{ }[\/latex]and[latex]\\text{ }\\left[0.1,0.5\\right].\\text{ }[\/latex]Determine the corresponding range for the viewing window. Show the graphs.<\/p>\n<\/div>\n

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

        \"Graph<\/p>\n

        window:[latex]\\text{ }\\left[-0.5,-0.1\\right];\\text{ }[\/latex]range:[latex]\\text{ }\\left[4,\\text{ }100\\right][\/latex]<\/p>\n

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

         <\/p>\n

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        \n

        window:[latex]\\text{ }\\left[0.1,\\text{ }0.5\\right];\\text{ }[\/latex]range:[latex]\\text{ }\\left[4,\\text{ }100\\right][\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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        56. Graph[latex]\\text{ }y=\\frac{1}{x}\\text{ }[\/latex]on the viewing window[latex]\\text{ }\\left[-0.5,-0.1\\right]\\text{ }[\/latex]and[latex]\\text{ }\\left[0.1,\\text{ }0.5\\right].\\text{ }[\/latex]Determine the corresponding range for the viewing window. Show the graphs.<\/p>\n<\/div>\n<\/div>\n<\/div>\n

        \n

        Extension<\/h4>\n
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        \n

        57. Suppose the range of a function[latex]\\text{ }f\\text{ }[\/latex]is[latex]\\text{ }\\left[-5,\\text{ }8\\right].\\text{ }[\/latex]What is the range of[latex]\\text{ }|f\\left(x\\right)|?[\/latex]<\/p>\n<\/div>\n

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

        [latex]\\left[0,\\text{ }8\\right][\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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        \n

        58. Create a function in which the range is all nonnegative real numbers.<\/p>\n<\/div>\n<\/div>\n

        \n
        \n

        59. Create a function in which the domain is[latex]\\text{ }x>2.[\/latex]<\/p>\n<\/div>\n

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

        Many answers. One function is[latex]\\text{ }f\\left(x\\right)=\\frac{1}{\\sqrt{x-2}}.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

        \n

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

        60. The height[latex]\\text{ }h\\text{ }[\/latex]of a projectile is a function of the time[latex]\\text{ }t\\text{ }[\/latex]it is in the air. The height in feet for[latex]\\text{ }t\\text{ }[\/latex]seconds is given by the function [latex]h\\left(t\\right)=-16{t}^{2}+96t.[\/latex] What is the domain of the function? What does the domain mean in the context of the problem?<\/p>\n<\/div>\n

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

        The domain is[latex]\\text{ }\\left[0,\\text{ }6\\right];\\text{ }[\/latex]it takes 6 seconds for the projectile to leave the ground and return to the ground<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

        \n
        \n

        61. The cost in dollars of making[latex]\\text{ }x\\text{ }[\/latex]items is given by the function[latex]\\text{ }C\\left(x\\right)=10x+500.[\/latex]<\/p>\n

          \n
        1. The fixed cost is determined when zero items are produced. Find the fixed cost for this item.<\/li>\n
        2. What is the cost of making 25 items?<\/li>\n
        3. Suppose the maximum cost allowed is $1500. What are the domain and range of the cost function,[latex]\\text{ }C\\left(x\\right)?[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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