Zeros of Rational Functions

In this section, you will use many of the same tools you used to find zeros of polynomials to find the zeros of rational functions. Finding the zeros (intercepts) will help you graph rational functions without a calculator.

INTERCEPTS OF RATIONAL FUNCTIONS

A rational function will have a $y$-intercept at $f(0)$, if the function is defined at zero. A rational function will not have a y-intercept if the function is not defined at zero.

Likewise, a rational function will have $x$-intercepts at the inputs that cause the output to be zero. Since a fraction is only equal to zero when the numerator is zero, $x$-intercepts can only occur when the numerator of the rational function is equal to zero.

EXAMPLE 10

Finding the Intercepts of a Rational Function

Find the intercepts of $f(x)=\frac{(x-2)(x+3)}{(x-1)(x+2)(x-5)}$.

Solution

We can find the $y$-intercept by evaluating the function at zero

\begin{aligned} f(0) &=\frac{(0-2)(0+3)}{(0-1)(0+2)(0-5)} \\ &=\frac{-6}{10} \\ &=-\frac{3}{5} \\ &=-0.6 \end{aligned}

The $x$-intercepts will occur when the function is equal to zero:

This is zero when the numerator is zero.

\begin{aligned} 0 &=\frac{(x-2)(x+3)}{(x-1)(x+2)(x-5)} \\ 0 &=(x-2)(x+3) \\ x &=2,-3 \end{aligned}

The $y$-intercept is $(0,–0.6)$, the $x$-intercepts are $(2,0)$ and $(–3,0)$. See Figure 16.

Figure 16

Source: Rice University, https://openstax.org/books/college-algebra/pages/5-6-rational-functions