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Mathematics LibreTexts

1.6.6E: Inverse Functions

  • Page ID
    30243
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    section 1.6 exercises

    Assume that the function f is a one-to-one function.

    1. If \(f(6)=7\) , find \(f^{-1} (7)\)

    2. If \(f(3)=2\) , find \(f^{-1} (2)\)

    3. If \(f^{-1} (-4)=-8\), find \(f(-8)\)

    4. If \(f^{-1} (-2)=-1\), find \(f(-1)\)

    5. If \(f(5)=2\), find \((f(5))^{-1}\)

    6. If \(f(1)=4\), find \((f(1))^{-1}\)

    7. Using the graph of \(f(x)\) shown

    屏幕快照 2019-06-17 下午5.06.51.png

    a. Find \(f(0)\)

    b. Solve \(f(x)=0\)

    c. Find \(f^{-1} (0)\)

    d. Solve \(f^{-1} (x)=0\)

    8. Using the graph shown

    屏幕快照 2019-06-17 下午5.08.02.png

    a. Find \(g(1)\)

    b. Solve \(g(x)=1\)

    c. Find \(g^{-1} (1)\)

    d. Solve \(g^{-1} (x)=1\)

    9. Use the table below to find the indicated quantities.

    \(x\) 0 1 2 3 4 5 6 7 8 9
    \(f(x)\) 8 0 7 4 2 6 5 3 9 1

    a. Find \(f(1)\)

    b. Solve \(f(x)=3\)

    c. Find \(f^{-1}(0)\)

    d. Solve \(f^{-1}(x)=7\)

    10. Use the table below to fill in the missing values.

    \(t\) 0 1 2 3 4 5 6 7 8
    \(h(t)\) 6 0 1 7 2 3 5 4 9

    a. Find \(h(6)\)

    b. Solve \(h(t)=0\)

    c. Find \(h^{-1} (5)\)

    d. Solve \(h^{-1} (t)=1\)

    For each table below, create a table for \(f^{-1} (x).\)

    11.

    \(x\) 3 6 9 13 14
    \(f(x)\) 1 4 7 12 16

    For each function below, find \(f^{-1} (x)\)

    13. \(f(x)=x+3\)

    14. \(f(x)=x+5\)

    15. \(f(x)= 2 - x\)

    16. \(f(x)=3-x\)

    17. \(f(x)=11x+7\)

    18. \(f(x)=9+10x\)

    For each function, find a domain on which \(f\) is one-to-one and non-decreasing, then find the inverse of \(f\) restricted to that domain.

    19. \(f(x)=(x +7)^{2}\)

    20. \(f(x)=(x-6)^{2}\)

    21. \(f(x)=x^{2} -5\)

    22. \(f(x)=x^{2} +1\)

    23. If \(f(x)=x^{3} -5\) and \(g(x)=\sqrt[{3}]{x+5}\), find

    a. \(f(g(x))\)

    b. \(g(f(x))\)

    c. What does this tell us about the relationship between \(f(x)\) and \(g(x)\)?

    24. If \(f(x)=\dfrac{x}{2+x}\) and \(g(x)=\dfrac{2x}{1-x}\), find

    a. \(f(g(x))\)

    b. \(g(f(x))\)

    c. What does this tell us about the relationship between \(f(x)\) and \(g(x)\)?

    Answer

    1. 6

    3. -4

    5. 1/2

    7a. 3
    b. 2
    c. 2
    d. 2

    11.

    \(x\) 1 4 7 12 16
    \(f^{-1}(x)\) 3 6 9 13 14

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

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

    17. \(f^{-1}(x) = \dfrac{x - 7}{11}\)

    19. Restricted domain \(x \ge -7\), \(f^{-1}(x) = \sqrt{x} - 7\)

    21. Restricted domain \(x \ge 0\), \(f^{-1}(x) = \sqrt{x + 5}\)

    23a. \(f(g(x)) = (\sqrt[3]{x + 5})^3 - 5 = x\)
    b. \(g(f(x)) = \sqrt[3]{x^3 - 5 + 5} = x\)
    c. This means that they are inverse functions (of each other)