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.

7 분 TEKS 2I 대수학 2

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.

Big idea

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

f(x) = 2x + 1 y = 2x + 1 (rewrite as y = …) x = 2y + 1 (swap x and y) x − 1 = 2y (isolate y) y = (x − 1) / 2 f⁻¹(x) = (x − 1) / 2

Practice

Find the inverse

If f(x) = 2x + 1, what is the inverse f⁻¹(x)?

Open the question →

The graph: reflect across y = x

The inverse is the mirror image of the original across the diagonal line y = x.

The horizontal-line test

Not every function has an inverse

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.

Why √x is "half" the inverse

√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.