Graph of an inverse function Consider a bijective function $f(x)$, i.e., a function that satisfies the horizontal line criterion and for which thus an inverse function $f^{-1}(x)$ exists. The graph of $f^{-1}$ arises from the graph of $f$ by reflection in the line with equation $y=x$.

In other words, the graphs of a function and its inverse are each other's mirror images with respect to reflection in the line $y=x$.

Constructing a function rule for an inverse function Construction of a function rule for the inverse function of a bijective function $f$ consists of two steps:

1. Simultaneously replace $x$ with $y$ and replace $y$ with $x$ in the equation $y=f(x)$. So you swap $x$ and $y$ with each other and get the equation $x=f(y)$.
2. Isolate $y$ in the equation $x=f(y)$. The right-hand side then becomes the function rule of $f^{-1}(x)$.

This is only possible for a function that satisfies the horizontal line criterion, i.e., for a function $f$ for which each horizontal line has at most one intersection with the graph of $f$.