Math
Inverse functions.
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 · 5 phases
Step by step.
- Replace f(x) with y.
- Swap x and y in the equation.
- Solve for y.
- Replace y with f⁻¹(x).
- Verify by checking that f(f⁻¹(x)) = x and f⁻¹(f(x)) = x.
Examples
Real-world.
- 1 If f(x) = 2x + 3, then f⁻¹(x) = (x − 3)/2
- 2 The inverse of f(x) = x³ is f⁻¹(x) = ∛x
- 3 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.