## Miscellaneous On-Line Topics forApplied CalculusFinite Mathematics & Applied CalculusExercises for Inverse Functions

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Find the range of each of the following functions.

1.           2.

3.           4.

5. $f(x) = \|x\| + 1$ 6. $g(x) = (x + 1)^2$

7.
 $h(x) = x + \frac{1}{x}$ with domain $(0, +∞)$
8. $f(x) = -x^3 -2x + 4$

In each of the following, decide whether the given function is one-to-one. In the event that a function is not one-to-one, specify two different values $x_1$ and $x_2$ of $x$ such that $f(x_1) = f(x_2).$
9.           10.

11.           12.

13. $f(x) = \|x\| + 1$ 14. $g(x) = (x + 1)^2,$ with domain $(0, +∞)$

15.
 $h(x) = x + \frac{1}{x}$ with domain $(0, +∞)$
16. $f(x) = x^3 - 4x^2 + 3x$

In each of the following, sketch the graph of the inverse of the given function.

 17. 18. 19. 20.

In the following exercises, verify that the given pairs of functions are inverse pairs. (Unless otherwise stated, all domains are the largest possible.)
21.
 $f(x) = 2x - 1$
22.
 $f(x) = x^{1/3}$
 $g(x) = \frac{x + 1}{2}$
 $g(x) = x^3$

23.
 $f(x) = (x - 1)^{1/3}$
24.
 $f(x) = \frac{x}{x + 1},$ with domain $[0, +∞)$
 $g(x) = x^3 + 1$
 $g(x) = \frac{x}{1 - x},$ with domain $[0, 1)$

25.
 $f(x) = \log_2(x+1),$ with domain $(0, +∞)$
26.
 $f(x) = 2 + e^{x-1}$ with domain all real numbers
 $g(x) = 2^x - 1,$ with domain all real numbers
 $g(x) = \ln(x-2) + 1$ with domain $(2, +∞)$

In each of the following exercises, find the inverse of the given function, specify the domains of the inverses, and check that your answer satisfies the definition for inverses. (Unless otherwise stated, all domains are the largest possible.)

27.
 $f(x) = 3x + 2$
 $f^{-1}(x) =$ The domain of $f^{-1}$ is: Select one All real numbers [0, +infinity) [1, +infinity) [2, +infinity) (-infinity, 1] (-infinity, 0] (-infinity, 2] (0, +infinity) (1, +infinity) (2, +infinity) [0, 10] (-infinity, 4] (0, 10)

28.
 $f(x) = \frac{2x - 1}{2}$
 $f^{-1}(x) =$ The domain of $f^{-1}$ is: Select one All real numbers [0, +infinity) [1, +infinity) [2, +infinity) (-infinity, 1] (-infinity, 0] (-infinity, 2] (0, +infinity) (1, +infinity) (2, +infinity) [0, 10] (-infinity, 4] (0, 10)

29.
 $f(x) = x$
 $f^{-1}(x) =$ The domain of $f^{-1}$ is: Select one All real numbers [0, +infinity) [1, +infinity) [2, +infinity) (-infinity, 1] (-infinity, 0] (-infinity, 2] (0, +infinity) (1, +infinity) (2, +infinity) [0, 10] (-infinity, 4] (0, 10)

30.
 $f(x) = (x + 1)^{1/2},$ with domain $[-1, +∞)$
 $f^{-1}(x) =$ The domain of $f^{-1}$ is: Select one All real numbers [0, +infinity) [1, +infinity) [2, +infinity) (-infinity, 1] (-infinity, 0] (-infinity, 2] (0, +infinity) (1, +infinity) (2, +infinity) [0, 10] (-infinity, 4] (0, 10)

31.
 $f(x) = (2x - 1)^3$
 $f^{-1}(x) =$ The domain of $f^{-1}$ is: Select one All real numbers [0, +infinity) [1, +infinity) [2, +infinity) (-infinity, 1] (-infinity, 0] (-infinity, 2] (0, +infinity) (1, +infinity) (2, +infinity) [0, 10] (-infinity, 4] (0, 10)

32.
 $f(x) = e^{4x+1}$
 $f^{-1}(x) =$ The domain of $f^{-1}$ is: Select one All real numbers [0, +infinity) [1, +infinity) [2, +infinity) (-infinity, 1] (-infinity, 0] (-infinity, 2] (0, +infinity) (1, +infinity) (2, +infinity) [0, 10] (-infinity, 4] (0, 10)

33.
 $f(x) = \log_3(x^3 + 1),$ with domain $(-1, +∞)$
 $f^{-1}(x) =$ The domain of $f^{-1}$ is: Select one All real numbers [0, +infinity) [1, +infinity) [2, +infinity) (-infinity, 1] (-infinity, 0] (-infinity, 2] (0, +infinity) (1, +infinity) (2, +infinity) [0, 10] (-infinity, 4] (0, 10)

34.
 $f(x) = 3(10)^{x+1}$
 $f^{-1}(x) =$ The domain of $f^{-1}$ is: Select one All real numbers [0, +infinity) [1, +infinity) [2, +infinity) (-infinity, 1] (-infinity, 0] (-infinity, 2] (0, +infinity) (1, +infinity) (2, +infinity) [0, 10] (-infinity, 4] (0, 10)

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Last Updated:June, 2001