Properties of the inverse and symmetry of the graph

The inverse “undoes” what the function did, and its graph is the reflection of that of $f$.

Property — Composition with the inverse

If $f^{-1}$ exists, then $f^{-1}\bigl(f(x)\bigr) = x \ \text{for every } x\in A, \qquad f\bigl(f^{-1}(y)\bigr) = y \ \text{for every } y\in B.$ That is: applying $f$ and then $f^{-1}$ (or vice versa) always returns the initial input.

Property — Graph of the inverse

The graph of $f^{-1}$ is obtained from the graph of $f$ by reflecting it in the line $y=x$: if $(a,b)$ is a point of the graph of $f$, then $(b,a)$ is a point of the graph of $f^{-1}$.

The graphs of $f(x)=\tfrac{x^2}{2}$ and of its inverse $f^{-1}(x)=\sqrt{2x}$ are symmetric with respect to the line $y=x$.

Links

Topics: Functions and properties
Concepts: Composite function · Inverse function
Skills: Interpreting a graph · Sketching a graph