Function Composition & Inverse

(f g)(x) = f(g(x)), and f^-1 undoes f — a reflection across y = x.

The idea

Definition (Composition).

For functions $f$ and $g$, the composition $f \circ g$ is the function $(f \circ g)(x) = f(g(x)),$ obtained by applying $g$ first and then $f$ to the result.

Read it from the inside out — $g$ acts first, though $f$ is written first. The order matters: squaring and then adding one is a different rule from adding one and then squaring.

Definition (Inverse function).

A function is the inverse of $f$, written $f^{-1}$, when applying it after $f$, or $f$ after it, returns every input to where it began: $f^{-1}(f(x)) = x \quad \text{and} \quad f(f^{-1}(x)) = x.$

The $-1$ is a name, not an exponent; $f^{-1}$ is not $1/f$. An inverse exists if and only if $f$ is one-to-one: if $f$ sent two inputs to one output, the inverse would have to send that output back to both, and no function may. To find $f^{-1}$, write $y = f(x)$ and solve for $x$ in terms of $y$.

Inverting swaps input with output. If $(a, b)$ lies on the graph of $f$, then $(b, a)$ lies on the graph of $f^{-1}$, so the two graphs are mirror images across the line $y = x$, and the domain of $f^{-1}$ is the range of $f$.

Ways to work on it

• Walkthrough. Compose, invert, the round-trip f^-1(f(x)) = x, and the reflection across y = x.
• Practice. Compose functions and find inverses.
• Hardest. Order of composition, restricted-domain inverses, and graph reflections.

Not sure where to start? Take the ten-question placement test.