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10.2E: Exercises

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    Practice Makes Perfect

    Exercise \(\PageIndex{19}\) Find and Evaluate Composite Functions

    In the following exercises, find

    1. \((f \circ g)(x)\)
    2. \((g \circ f)(x)\)
    3. \((f \cdot g)(x)\)
    1. \(f(x)=4 x+3\) and \(g(x)=2 x+5\)
    2. \(f(x)=3 x-1\) and \(g(x)=5 x-3\)
    3. \(f(x)=6 x-5\) and \(g(x)=4 x+1\)
    4. \(f(x)=2 x+7\) and \(g(x)=3 x-4\)
    5. \(f(x)=3 x\) and \(g(x)=2 x^{2}-3 x\)
    6. \(f(x)=2 x\) and \(g(x)=3 x^{2}-1\)
    7. \(f(x)=2 x-1\) and \(g(x)=x^{2}+2\)
    8. \(f(x)=4 x+3\) and \(g(x)=x^{2}-4\)
    Answer

    1.

    1. \(8x+23\)
    2. \(8x+11\)
    3. \(8 x^{2}+26 x+15\)

    3.

    1. \(24x+1\)
    2. \(24x-19\)
    3. \(24x^{2}+19x-5\)

    5.

    1. \(6 x^{2}-9 x\)
    2. \(18 x^{2}-9 x\)
    3. \(6 x^{3}-9 x^{2}\)

    7.

    1. \(2 x^{2}+3\)
    2. \(4 x^{2}-4 x+3\)
    3. \(2 x^{3}-x^{2}+4 x-2\)
    Exercise \(\PageIndex{20}\) Find and Evaluate Composite Functions

    In the following exercises, find the values described.

    1. For functions \(f(x)=2 x^{2}+3\) and \(g(x)=5x-1\), find
      1. \((f \circ g)(-2)\)
      2. \((g \circ f)(-3)\)
      3. \((f \circ f)(-1)\)
    2. For functions \(f(x)=5 x^{2}-1\) and \(g(x)=4x−1\), find
      1. \((f \circ g)(1)\)
      2. \((g \circ f)(-1)\)
      3. \((f \circ f)(2)\)
    3. For functions \(f(x)=2x^{3}\) and \(g(x)=3x^{2}+2\), find
      1. \((f \circ g)(-1)\)
      2. \((g \circ f)(1)\)
      3. \((g \circ g)(1)\)
    4. For functions \(f(x)=3 x^{3}+1\) and \(g(x)=2 x^{2}=3\), find
      1. \((f \circ g)(-2)\)
      2. \((g \circ f)(-1)\)
      3. \((g \circ g)(1)\)
    Answer

    1.

    1. \(245\)
    2. \(104\)
    3. \(53\)

    3.

    1. \(250\)
    2. \(14\)
    3. \(77\)
    Exercise \(\PageIndex{21}\) Determine Whether a Function is One-to-One

    In the following exercises, determine if the set of ordered pairs represents a function and if so, is the function one-to-one.

    1. \(\begin{array}{l}{\{(-3,9),(-2,4),(-1,1),(0,0)}, {(1,1),(2,4),(3,9) \}}\end{array}\)
    2. \(\begin{array}{l}{\{(9,-3),(4,-2),(1,-1),(0,0)}, {(1,1),(4,2),(9,3) \}}\end{array}\)
    3. \(\begin{array}{l}{\{(-3,-5),(-2,-3),(-1,-1)}, {(0,1),(1,3),(2,5),(3,7) \}}\end{array}\)
    4. \(\begin{array}{l}{\{(5,3),(4,2),(3,1),(2,0)}, {(1,-1),(0,-2),(-1,-3) \}}\end{array}\)
    Answer

    1. Function; not one-to-one

    3. One-to-one function

    Exercise \(\PageIndex{22}\) Determine Whether a Function is One-to-One

    In the following exercises, determine whether each graph is the graph of a function and if so, is it one-to-one.

    1.


    1. This figure shows a graph of a circle with center at the origin and radius 3.
      Figure 10.1.65

    2. This figure shows a graph of a parabola opening upward with vertex at (0k, 2).
      Figure 10.1.66

    2.


    1. This figure shows a parabola opening to the right with vertex at (negative 2, 0).
      Figure 10.1.67

    2. This figure shows a graph of a polynomial with odd order, so that it starts in the third quadrant, increases to the origin and then continues increasing through the first quadrant.
      Figure 10.1.68

    3.


    1. This figure shows a graph of a curve that starts at (negative 6 negative 2) increases to the origin and then continues increasing slowly to (6, 2).
      Figure 10.1.69

    2. This figure shows a parabola opening upward with vertex at (0, negative 4).
      Figure 10.1.70

    4.


    1. This figure shows a straight line segment decreasing from (negative 4, 6) to (2, 0), after which it increases from (2, 0) to (6, 4).
      Figure 10.1.71

    2. This figure shows a circle with radius 4 and center at the origin.
      Figure 10.1.72
    Answer

    1.

    1. Not a function
    2. Function; not one-to-one

    3.

    1. One-to-one function
    2. Function; not one-to-one
    Exercise \(\PageIndex{23}\) Determine Whether a Function is One-to-One

    In the following exercises, find the inverse of each function. Determine the domain and range of the inverse function.

    1. \(\{(2,1),(4,2),(6,3),(8,4)\}\)
    2. \(\{(6,2),(9,5),(12,8),(15,11)\}\)
    3. \(\{(0,-2),(1,3),(2,7),(3,12)\}\)
    4. \(\{(0,0),(1,1),(2,4),(3,9)\}\)
    5. \(\{(-2,-3),(-1,-1),(0,1),(1,3)\}\)
    6. \(\{(5,3),(4,2),(3,1),(2,0)\}\)
    Answer

    1. \(\begin{array}{l}{\text { Inverse function: }\{(1,2),(2,4),(3,6),(4,8)\} . \text { Domain: }\{1,2,3,4\} . \text { Range: }} {\{2,4,6,8\} .}\end{array}\)

    3. \(\begin{array}{l}{\text { Inverse function: }\{(-2,0),(3,1),(7,2),(12,3)\} . \text { Domain: }\{-2,3,7,12\} \text { . }} {\text { Range: }\{0,1,2,3\}}\end{array}\)

    5. \(\begin{array}{l}{\text { Inverse function: }\{(-3,-2),(-1,-1),(1,0),(3,1)\} . \text { Domain: }} {\{-3,-1,1,3\} . \text { Range: }\{-2,-1,0,1\}}\end{array}\)

    Exercise \(\PageIndex{24}\) Determine Whether a Function is One-to-One

    In the following exercises, graph, on the same coordinate system, the inverse of the one-to-one function shown.


    1. This figure shows a series of line segments from (negative 4, negative 3) to (negative 3, 0) then to (negative 1, 2) and then to (3, 4).
      Figure 10.1.73

    2. This figure shows a series of line segments from (negative 4, negative 4) to (negative 3, 1) then to (0, 2) and then to (2, 4).
      Figure 10.1.74

    3. This figure shows a series of line segments from (negative 4, 4) to (0, 3) then to (3, 2) and then to (4, negative 1).
      Figure 10.1.75

    4. This figure shows a series of line segments from (negative 4, negative 4) to (negative 1, negative 3) then to (0, 1), then to (1, 3), and then to (4, 4).
      Figure 10.1.76
    Answer

    1.

    This figure shows a series of line segments from (negative 3, negative 4) to (0, negative 3) then to (2, negative 1), and then to (4, 3).
    Figure 10.1.77

    3.

    This figure shows a series of line segments from (negative 1, 4) to (2, 3) then to (3, 0), and then to (4, negative 4).
    Figure 10.1.78
    Exercise \(\PageIndex{25}\) Determine Whether the given functions are inverses

    In the following exercises, determine whether or not the given functions are inverses.

    1. \(f(x)=x+8\) and \(g(x)=x-8\)
    2. \(f(x)=x-9\) and \(g(x)=x+9\)
    3. \(f(x)=7 x\) and \(g(x)=\frac{x}{7}\)
    4. \(f(x)=\frac{x}{11}\) and \(g(x)=11 x\)
    5. \(f(x)=7 x+3\) and \(g(x)=\frac{x-3}{7}\)
    6. \(f(x)=5 x-4\) and \(g(x)=\frac{x-4}{5}\)
    7. \(f(x)=\sqrt{x+2}\) and \(g(x)=x^{2}-2\)
    8. \(f(x)=\sqrt[3]{x-4}\) and \(g(x)=x^{3}+4\)
    Answer

    1. \(g(f(x))=x,\) and \(f(g(x))=x,\) so they are inverses.

    3. \(g(f(x))=x,\) and \(f(g(x))=x,\) so they are inverses.

    5. \(g(f(x))=x,\) and \(f(g(x))=x,\) so they are inverses.

    7. \(g(f(x))=x,\) and \(f(g(x))=x,\) so they are inverses (for nonnegative \(x )\)

    Exercise \(\PageIndex{26}\) Determine the inverse of a function

    In the following exercises, find the inverse of each function.

    1. \(f(x)=x-12\)
    2. \(f(x)=x+17\)
    3. \(f(x)=9 x\)
    4. \(f(x)=8 x\)
    5. \(f(x)=\frac{x}{6}\)
    6. \(f(x)=\frac{x}{4}\)
    7. \(f(x)=6 x-7\)
    8. \(f(x)=7 x-1\)
    9. \(f(x)=-2 x+5\)
    10. \(f(x)=-5 x-4\)
    11. \(f(x)=x^{2}+6, x \geq 0\)
    12. \(f(x)=x^{2}-9, x \geq 0\)
    13. \(f(x)=x^{3}-4\)
    14. \(f(x)=x^{3}+6\)
    15. \(f(x)=\frac{1}{x+2}\)
    16. \(f(x)=\frac{1}{x-6}\)
    17. \(f(x)=\sqrt{x-2}, x \geq 2\)
    18. \(f(x)=\sqrt{x+8}, x \geq-8\)
    19. \(f(x)=\sqrt[3]{x-3}\)
    20. \(f(x)=\sqrt[3]{x+5}\)
    21. \(f(x)=\sqrt[4]{9 x-5}, x \geq \frac{5}{9}\)
    22. \(f(x)=\sqrt[4]{8 x-3}, x \geq \frac{3}{8}\)
    23. \(f(x)=\sqrt[5]{-3 x+5}\)
    24. \(f(x)=\sqrt[5]{-4 x-3}\)
    Answer

    1. \(f^{-1}(x)=x+12\)

    3. \(f^{-1}(x)=\frac{x}{9}\)

    5. \(f^{-1}(x)=6 x\)

    7. \(f^{-1}(x)=\frac{x+7}{6}\)

    9. \(f^{-1}(x)=\frac{x-5}{-2}\)

    11. \(f^{-1}(x)=\sqrt{x-6}\)

    13. \(f^{-1}(x)=\sqrt[3]{x+4}\)

    15. \(f^{-1}(x)=\frac{1}{x}-2\)

    17. \(f^{-1}(x)=x^{2}+2, x \geq 0\)

    19. \(f^{-1}(x)=x^{3}+3\)

    21. \(f^{-1}(x)=\frac{x^{4}+5}{9}, x \geq 0\)

    23. \(f^{-1}(x)=\frac{x^{5}-5}{-3}\)

    Exercise \(\PageIndex{27}\) Writing Exercises
    1. Explain how the graph of the inverse of a function is related to the graph of the function.
    2. Explain how to find the inverse of a function from its equation. Use an example to demonstrate the steps.
    Answer

    1. Answers will vary.

    Self Check

    a. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

    This table has four rows and four columns. The first row, which serves as a header, reads I can…, Confidently, With some help, and No—I don’t get it. The first column below the header row reads Find and evaluate composite functions, determine whether a function is one-to-one, and find the inverse of a function. The rest of the cells are blank.
    Figure 10.1.79

    b. If most of your checks were:

    …confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.

    …with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

    …no—I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.

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