Inverse Function Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Inverse Function.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept 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.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: f^{-1}(f(x)) = x and f(f^{-1}(x)) = x. They undo each other.

Common stuck point: f^{-1}(x) is NOT \frac{1}{f(x)}. The -1 means inverse, not reciprocal.

Sense of Study hint: Write y = f(x), then swap x and y, and solve the new equation for y. That gives you the inverse.

Worked Examples

Example 1

easy
Find the inverse of f(x) = 3x - 7.

Solution

  1. 1
    Write y = 3x - 7.
  2. 2
    Swap x and y: x = 3y - 7.
  3. 3
    Solve for y: 3y = x + 7, so y = \frac{x + 7}{3}.
  4. 4
    Therefore f^{-1}(x) = \frac{x + 7}{3}.

Answer

f^{-1}(x) = \frac{x + 7}{3}
To find an inverse, swap input and output then solve for the new output. The inverse 'undoes' the original function: applying f then f^{-1} returns the original input.

Example 2

medium
Find the inverse of f(x) = \frac{2x + 5}{x - 1} for x \neq 1.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Find the inverse of f(x) = \frac{x}{4} + 2.

Example 2

hard
If f(x) = 2^x, find f^{-1}(x). Then verify by showing that f(f^{-1}(8)) = 8.
← Back to Inverse Function Formula Explained Practice Mode

Background Knowledge

These ideas may be useful before you work through the harder examples.

function definition