Inverse function

Definition

Let $f \colon A \to B$ be a bijective function. The inverse function of $f$ is the function $f^{-1} \colon B \to A$ such that, for each $b \in B$, $f^{-1}(b)$ is the unique $a \in A$ with $f(a) = b$.

Remarks

Then $f^{-1} \circ f = \mathrm{id}_A$ and $f \circ f^{-1} = \mathrm{id}_B$. See bijective iff it has a two-sided inverse. The inverse $f^{-1}$ is a bijection, and its inverse is $f$. Example: the inverse of $x \mapsto 2x + 1$ on $\mathbb{R}$ is $y \mapsto (y - 1)/2$. Do not confuse $f^{-1}(b)$ with the preimage $f^{-1}(\{b\})$, which exists for every function.

Explanations

L1 human

To find the inverse of $y = 2x + 1$, solve for $x$: $x = (y - 1)/2$. The inverse runs the function backwards. Its graph is the graph of $f$ reflected in the line $y = x$.