# 1.9 | Inverse Functions

Let $$f$$ and $$g$$ be two functions such that
$$f(g(x)) = x,$$
for every $$x$$ in the domain of $$g$$ and
$$g(f(x)) = x,$$
for every $$x$$ in the domain of $$f.$$
Under these conditions, the function $$g$$ is called the inverse function of the function $$f.$$ The function $$g$$ is denoted by $$f^{-1}$$ (read “f-inverse”). So,
$$f(f^{-1}(x)) = x$$
and
$$f^{-1}(f(x)) = x.$$
The domain of $$f$$ must be equal to the range of $$f^{-1},$$ and the range of $$f$$ must be equal to the domain of $$f^{-1}.$$
Horizontal Line Test A function $$f$$ has an inverse function if and only if no horizontal line intersects the graph of $$f$$ at more than one point.
A function $$f$$ is one-to-one if each value of the dependent variable corresponds to exactly one of the independent variable. A function $$f$$ has an inverse function if and only if $$f$$ is one-to-one.
Finding an Inverse Function

1. Use the Horizontal Line Test to decide whether $$f$$ has an inverse function.
2. In the equation for $$f(x),$$ replace $$f(x)$$ by $$y.$$
3. Interchange the roles of $$x$$ and $$y,$$ and solve for $$y.$$
4. Replace $$y$$ by $$f^{-1}(x)$$ in the new equation.
5. Verify that $$f$$ and $$f^{-1}$$ are inverse functions of each other by showing that the domain of $$f$$ is equal to the range of $$f^{-1}$$, the range of $$f$$ is equal to the domain of $$f^{-1},$$ and $$f(f^{-1}(x)) = x$$ and $$f^{-1}(f(x)) = x.$$