Practice Inverse Function in Math

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

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Quick Recap

The inverse of a function f is a function f−1 that reverses f: if f(a)=b then f−1(b)=a. It exists only when f is one-to-one.

If f turns a into b, then f−1 turns b back into a. Reverse the process.

Showing a random 20 of 50 problems.

Example 1

medium
If f(x)=e3x, find f−1(x).

Example 2

hard
For f(x)=xx+1 on R∖{−1}, find f−1 and its domain.

Example 3

easy
Is f(x)=x2 on all reals invertible?

Example 4

challenge
A linear function satisfies f(x)=f−1(x) for all x and f(x)=ax+b with a≠1. Find a.

Example 5

medium
On what domain is f(x)=x2 invertible, and what is the inverse?

Example 6

medium
If f(x)=log⁡3x, find f−1(x).

Example 7

easy
If f−1(x)=x−2, find f(x).

Example 8

hard
If f(x)=x3+1, find (f−1)′(2). (Use the rule (f−1)′(b)=1/f′(a) where f(a)=b.)

Example 9

medium
Find f−1(x) for f(x)=ln⁡(x−2).

Example 10

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

Example 11

medium
For f(x)=6−2x, solve f−1(x)=4.

Example 12

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

Example 13

medium
Two functions are graphed: y=f(x) and y=f−1(x). What is the relationship between the two graphs?

Example 14

easy
If f(2)=7, what is f−1(7)?

Example 15

easy
True or False: f−1(x) means 1/f(x).

Example 16

hard
If f(x)=2x, find f−1(x). Then verify by showing that f(f−1(8))=8.

Example 17

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

Example 18

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

Example 19

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

Example 20

easy
Find the inverse of f(x)=3x−7.
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