Reflection in the line y = x

Functions and graphs: Transformations of graphs and functions

Reflection in the line y = x

There is one more special transformation of the graph of a function, which is mirroring in the oblique line with equation \(y=x\). This transformation swaps the coordinates of a point. This line mirroring fits the creation of the graph of an inverse function \(f^{-1}\) given the invertible function \(f\).

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 \(y=x\).

If \(\bigl(x,f(x)\bigr)\) is a point on the graph of \(f\), then \(\bigl(f(x),x\bigr)\) is a point on the graph of \(f^{-1}\). After all, by definition of an inverse function we have: \(f^{-1}\bigl(f(x)\bigr)=x\)

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:

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 \(x=f(y)\).
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 \(f\).