Practice Inverse Function in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

Worksheet PDF Free Answer-key PDF Family

Quick Recap

The inverse of a function ff is a function fβˆ’1f^{-1} that reverses ff: if f(a)=bf(a) = b then fβˆ’1(b)=af^{-1}(b) = a. It exists only when ff is one-to-one.

If ff turns aa into bb, then fβˆ’1f^{-1} turns bb back into aa. Reverse the process.

Showing a random 20 of 50 problems.

Example 1

medium
If f(x)=e3xf(x) = e^{3x}, find fβˆ’1(x)f^{-1}(x).

Example 2

hard
For f(x)=xx+1f(x) = \frac{x}{x+1} on Rβˆ–{βˆ’1}\mathbb{R} \setminus \{-1\}, find fβˆ’1f^{-1} and its domain.

Example 3

easy
Is f(x)=x2f(x)=x^2 on all reals invertible?

Example 4

challenge
A linear function satisfies f(x)=fβˆ’1(x)f(x)=f^{-1}(x) for all xx and f(x)=ax+bf(x)=ax+b with aβ‰ 1a\neq 1. Find aa.

Example 5

medium
On what domain is f(x)=x2f(x) = x^2 invertible, and what is the inverse?

Example 6

medium
If f(x)=log⁑3xf(x)=\log_3 x, find fβˆ’1(x)f^{-1}(x).

Example 7

easy
If fβˆ’1(x)=xβˆ’2f^{-1}(x)=x-2, find f(x)f(x).

Example 8

hard
If f(x)=x3+1f(x) = x^3 + 1, find (fβˆ’1)β€²(2)(f^{-1})'(2). (Use the rule (fβˆ’1)β€²(b)=1/fβ€²(a)(f^{-1})'(b) = 1/f'(a) where f(a)=bf(a)=b.)

Example 9

medium
Find fβˆ’1(x)f^{-1}(x) for f(x)=ln⁑(xβˆ’2)f(x) = \ln(x - 2).

Example 10

easy
Is f(x)=∣x∣f(x) = |x| invertible on all real numbers?

Example 11

medium
For f(x)=6βˆ’2xf(x) = 6 - 2x, solve fβˆ’1(x)=4f^{-1}(x) = 4.

Example 12

medium
If ff and gg are inverses and g(5)=3g(5)=3, find f(3)f(3).

Example 13

medium
Two functions are graphed: y=f(x)y = f(x) and y=fβˆ’1(x)y = f^{-1}(x). What is the relationship between the two graphs?

Example 14

easy
If f(2)=7f(2)=7, what is fβˆ’1(7)f^{-1}(7)?

Example 15

easy
True or False: fβˆ’1(x)f^{-1}(x) means 1/f(x)1/f(x).

Example 16

hard
If f(x)=2xf(x) = 2^x, find fβˆ’1(x)f^{-1}(x). Then verify by showing that f(fβˆ’1(8))=8f(f^{-1}(8)) = 8.

Example 17

easy
Find the inverse of f(x)=3xf(x)=3x.

Example 18

challenge
If f(g(x))=xf(g(x))=x for all xx and f(x)=3x+2f(x)=3x+2, find g(x)g(x).

Example 19

medium
Find the inverse of f(x)=2xf(x)=2^x.

Example 20

easy
Find the inverse of f(x)=3xβˆ’7f(x) = 3x - 7.
← Back to Inverse Function Worked Examples Formula Explained