2.8 Inverse Functions

What Makes a Function Invertible

A function has an inverse function only if it is one-to-one (passes the horizontal line test) — each output comes from exactly one input. If f is not one-to-one on its whole domain, we can often restrict the domain to a piece where it is, then define the inverse on that restricted piece.

Finding and Interpreting f⁻¹

The inverse swaps inputs and outputs: if f(a) = b, then f⁻¹(b) = a. To find f⁻¹ algebraically, swap x and y in y = f(x), then solve for y. Domain of f⁻¹ = range of f, and range of f⁻¹ = domain of f. Composing gives f(f⁻¹(x)) = x and f⁻¹(f(x)) = x.

Graphs and Tables of Inverses

The graph of f⁻¹ is the reflection of f across the line y = x. Points (a, b) on f become (b, a) on f⁻¹. In table form, the input and output rows/columns simply switch places. Increasing functions have increasing inverses; decreasing functions have decreasing inverses.

Key terms

One-to-one function

A function where each output corresponds to exactly one input; passes the horizontal line test.

Inverse function notation

f⁻¹(x); if f(a) = b then f⁻¹(b) = a.

Domain/Range relationship for inverses

Domain of f⁻¹ = range of f; range of f⁻¹ = domain of f.

Graphical relationship of f and f⁻¹

The graph of f⁻¹ is the reflection of the graph of f across the line y = x.

Composition property of inverses

f(f⁻¹(x)) = x and f⁻¹(f(x)) = x for all x in the appropriate domains.

Try an AP-style question

What is the inverse of f(x) = 2x + 3?

1. (x + 3)/2
2. 2x − 3
3. (x − 3)/2
4. 1/(2x + 3)