Given a function $f\left(x\right)$, the inverse of $f\left(x\right)$ is the function $g\left(x\right)$ which has the property that $y\=g\left(x\right)$ exactly when $x\=f\left(y\right)$ (for the same values of $x$ and $y$). That is, the inverse of a function exactly undoes whatever the function does. The inverse of the function $f\left(x\right)$ is commonly denoted by ${f}^{1}\left(x\right)$.
Some functions have inverses (they are called invertible) and some do not (they are called noninvertible). An easy way to tell if a function is invertible or not is whether or not it passes the horizontal line test:
If any horizontal line passes through more than one point on the graph of y=f(x), the function f(x) is not invertible.
If no such horizontal line exists, the function is invertible.
Even if $f\left(x\right)$ is not invertible, it might still have a partial inverse. If you restrict the domain of $f\left(x\right)$, creating a new function $g\left(x\right)$ which does pass the horizontal line test, then $g\left(x\right)$ is invertible, and its inverse is called a partial inverse of $f\left(x\right)$.
In order to graph a function's inverse, simply reflect its graph through the line $yequals;x$.
Note : Even though they look similar, the inverse of $f\left(x\right)$, denoted ${f}^{1}\left(x\right)$, is not the same as the reciprocal of $f\left(x\right)$, which can be written $f{\left(x\right)}^{1}$. For example, the inverse of the function $f\left(x\right)\={x}^{2}$ is the function ${f}^{1}\left(x\right)\=\sqrt{x}$ , not the expression $\frac{1}{\sqrt{x}}$. The notation ${f}^{1}$ is intended to represent the concept of "inverting the action of $f\left(x\right)$", not "inverting the result of $f\left(x\right)$".
