Inverse functions

Math

Definition

A function that reverses the operation of another function. If f takes input a to output b, then f⁻¹ takes b back to a. A function must be one-to-one (pass the horizontal line test) to have an inverse.

How It Works

  1. Replace f(x) with y.
  2. Swap x and y in the equation.
  3. Solve for y.
  4. Replace y with f⁻¹(x).
  5. Verify by checking that f(f⁻¹(x)) = x and f⁻¹(f(x)) = x.

Examples

  • If f(x) = 2x + 3, then f⁻¹(x) = (x − 3)/2
  • The inverse of f(x) = x³ is f⁻¹(x) = ∛x
  • Converting Celsius to Fahrenheit and back are inverse operations
Key Fact

f(f⁻¹(x)) = x and f⁻¹(f(x)) = x; the graph of f⁻¹ is the reflection of f over y = x.

Related Terms

Function notation Composition of functions Domain and range

Study This Concept

Practice inverse functions with free review games in these units:

AP Calculus AB Unit 3: Differentiation: Composite, Implicit