Introduction: Sometimes we know the function $y(x)$, but are we interested in the function $x(y)$. This function $x(y)$ is then the inverse of the function $y(x)$.
Definition: IIf $y(x)$ is a function of the variable $x$ and $x(y)$ is a function of the variable $y$ satisfying the property
\[y(x(y)) = y \qquad \text{and} \qquad x(y(x)) = x,\]
then $x(y)$ is called the inverse function of the function $y(x)$.
Remark: One can see this graphically as well. For a function $y(x)$ one starts at the $x$-axis and reads, via the graph of $y(x)$ the corresponding value on the $y$-axis. This is shown in the left figure below. If you consider the inverse functions of $y(x)$, the function $x(y)$, then you start at the $y$-axis and read, via de graph, the corresponding value on the $x$-axis.
Definition: IIf $y(x)$ is a function of the variable $x$ and $x(y)$ is a function of the variable $y$ satisfying the property
\[y(x(y)) = y \qquad \text{and} \qquad x(y(x)) = x,\]
then $x(y)$ is called the inverse function of the function $y(x)$.
Remark: One can see this graphically as well. For a function $y(x)$ one starts at the $x$-axis and reads, via the graph of $y(x)$ the corresponding value on the $y$-axis. This is shown in the left figure below. If you consider the inverse functions of $y(x)$, the function $x(y)$, then you start at the $y$-axis and read, via de graph, the corresponding value on the $x$-axis.
