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Inverse Functions Basics

A free Algebra II lesson from the “Functions and Notation” unit, with a worked example and practice problems including step-by-step solutions.

An inverse function reverses the action of another function. If a function sends an input to an output, the inverse sends that output back to the original input. This idea matters because inverses help solve equations and connect operations such as squaring and square rooting, or exponentials and logarithms. When practicing, think about undoing operations in the opposite order. For symbolic inverses, replace f(x) with y, switch x and y, then solve for y. A common mistake is reversing only one operation instead of undoing the whole rule step by step.

What you'll learn

Why it matters: Converting Celsius to Fahrenheit and back, decoding a coded message, and finding the pre-tax price on a receipt are inverse-function moves. The inverse undoes the original rule's steps in reverse order, which is also why squaring and square-rooting, or exponentials and logarithms, travel as paired operations.

Worked example

Problem. Find the inverse of f(x) = 2x + 5.

  1. Write y = 2x + 5.
  2. Swap x and y: x = 2y + 5.
  3. Solve: y = (x - 5)/2.

Answer: f^-1(x) = (x - 5)/2

Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (typed response and multiple choice). Counts include repeated prompts and variants.

Practice problems

1. If f(x) = x + 9, find f^-1(14).

Show solution
  1. The inverse undoes +9.
  2. 14 - 9 = 5.

Answer: 5

2. If f(x) = 3x, find f^-1(21).

Show solution
  1. Undo multiplying by 3.

Answer: 7

3. The inverse of f(x) = x - 4 is...

Choices: f^-1(x) = x + 4 · f^-1(x) = x - 4 · f^-1(x) = 4x · f^-1(x) = x/4

Show solution
  1. Undo subtracting 4 by adding 4.

Answer: f^-1(x) = x + 4

4. The inverse of f(x) = 5x is...

Choices: f^-1(x) = x/5 · f^-1(x) = x - 5 · f^-1(x) = x + 5 · f^-1(x) = 5x

Show solution
  1. Undo multiplying by 5.

Answer: f^-1(x) = x/5

5. The inverse of f(x) = 2x - 6 is...

Choices: f^-1(x) = (x + 6)/2 · f^-1(x) = (x - 6)/2 · f^-1(x) = 2x + 6 · f^-1(x) = x/2 - 6

Show solution
  1. Swap x and y, then solve.

Answer: f^-1(x) = (x + 6)/2

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