Inverse Functions: How to Reverse a Rule
An inverse function undoes the original. Swap x and y, solve for y — that's the whole technique. Plus the geometric meaning (reflection across y = x) and the horizontal-line test for invertibility.
An inverse undoes the original
If a function f sends 3 → 7, the inverse f⁻¹ sends 7 → 3. Whatever rule f applies, f⁻¹ reverses it step by step. Composition gives back the original input: f⁻¹(f(x)) = x.
The inverse swaps the roles of input and output. Algebraically: switch x and y, then solve for y. Graphically: reflect across the line y = x.
The swap-and-solve method
The graph: reflect across y = x
The horizontal-line test
For the inverse to also be a function, the original must pass the horizontal-line test: no horizontal line crosses the graph more than once. Functions that fail (like y = x²) need their domain restricted before inverting.
Example: restricting a quadratic
f(x) = x² over all reals fails the horizontal-line test (y = 4 hits twice). But if we restrict to x ≥ 0, the inverse is f⁻¹(x) = √x.
√4 = 2 (positive root only) — that's why the inverse of x² (restricted to x ≥ 0) is √x, not ±√x. Mathematicians chose the positive branch as the principal inverse.
3-second recap
- f(f⁻¹(x)) = x and f⁻¹(f(x)) = x — they undo each other.
- To find: swap x and y, solve for y.
- To graph: reflect across y = x.
- Horizontal-line test fails → restrict domain first.