Zeros 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
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