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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