In chapter 3, we discussed that every function has an inverse, but only a one-to-one function has an inverse function. Some rational functions are one-to-one functions such as $f(x)=\dfrac{1}{x}$ or $f(x)=\dfrac{x-1}{x+4}$. Therefore, their inverse is a function. Some rational functions are many-to-one functions such as $f(x)=\dfrac{1}{x^2-1}$ or $f(x)=\dfrac{x^3-4x^2+2}{x^2-1}$. Therefore, their inverse is not a function (Figure 1).

One-to-one rational functions that have an inverse function	Many-to-one rational functions that do not have an inverse function
Figure 1. Graphs of rational functions

Notice that three of the graphs in figure 1 have horizontal and vertical asymptotes but the 4th graph has two vertical asymptotes and a slant asymptote. A slant asymptote is a line of the form $y=mx+b$ that is neither vertical nor horizontal but that the graph gets closer and closer to as $x$ approaches positive and negative infinity. Slant asymptotes occur in the graph of a rational function $f(x)=\dfrac{P(x)}{Q(x)}$ when the degree of $P(x)$ is one more than the degree of $Q(x)$. For example, in the function $f(x)=\dfrac{x^3-4x^2+2}{x^2-1}$ (Figure 1), $P(x)=x^3-4x^2+2$ and has degree 3, while $Q(x)=x^2-1$ which has degree 2. Since 3 is one more than 2, there is a slant asymptote.

Graphing the Inverse Function of a Rational Function

We may graph the inverse of a rational function by creating and using its inverse table. For example, given the function $f(x)=\dfrac{1}{x}$, we may graph the function by creating a table of values (Table 1).

The inverse of the function is found by switching the values of the $x$ and $y$ columns so that the inputs become the values of $y$ and the outputs become the values of $x$. The table after switching the values of the $x$ and $y$ columns is the inverse table (Table 2).

Figure 2 shows the graph of the inverse of the function $f(x)=\dfrac{1}{x}$ drawn from its inverse table. Notice that the graph of the inverse is exactly the same as the graph of the original function $f(x)=\dfrac{1}{x}$. In other words, the function $f(x)=\dfrac{1}{x}$ is the inverse of the function itself. The inverse function is a reflection of the original function with respect to the line of symmetry $y=x$.

Figure 2. The inverse of the function $f(x)=\dfrac{1}{x}$ is $f^{-1}(x)=\dfrac{1}{x}$.

Since the graph of the function $f(x)=\dfrac{1}{x}$ is symmetric across the line $y=x$, the inverse function is identical to the original function.

Determining the Inverse Function of an One-to-One Rational Function

To determine the equation of the inverse function of a one-to-one rational function, we use the same idea of switching the input and output. We start by writing $y=f(x)$, switch $x$ and $y$, and then solve for $y$.

For example, to determine the inverse function of the one-to-one rational function $g(x)=\dfrac{1}{x}$, we write $y=g(x)$ then switch $x$ and $y$:

$\begin{aligned}g(x)&=\dfrac{1}{x}\\\\y&=\dfrac{1}{x}\\\\x&=\dfrac{1}{y}\end{aligned}$

At this point, we have the inverse. We now need to solve for $y$ so we can write the inverse using function notation by replacing $y$ with $g^{-1}(x)$:

$\begin{aligned}x&=\dfrac{1}{y}\\\\x\color{blue}{\cdot y}&=\dfrac{1}{y}\color{blue}{\cdot y}&&\text{Multiply both sides by }y\text{ to clear the fractions}\\\\xy&=1\\\\y&=\dfrac{1}{x}\\\\g^{-1}(x)&=\dfrac{1}{x}\end{aligned}$

Therefore, the equation of the inverse function is $g^{-1}(x)=\dfrac{1}{x}$.