Chapter 3.7, Problem 15E

To determine

To find the $x$ -intercept and $y$ -intercept of the rational function.

Answer to Problem 15E

The $x$ -intercept are $3\text{ and }-3$ and there is no $y$ -intercept of the given rational function.

Explanation of Solution

Given information:

The rational function is $r\left(x\right)=\frac{x^2-9}{x^2}$

Calculation:

To find $x$ -intercept of the function, determine zeros of the numerator,

Therefore,

  $\begin{array}{l}{x}^2-9=0\\ {\left(x\right)}^2-{\left(3\right)}^2=0\\ \left(x+3\right)\left(x-3\right)=0\end{array}$

Either,

  $\begin{array}{l}x+3=0\\ x=-3\end{array}$

Or,

  $\begin{array}{l}x-3=0\\ x=3\end{array}$

So, the $x$ -interceptsof the given rational function are $3\text{ and }-3$

To find $y$ -intercept of the function, substitute, $x=0$ into the original form of the function,

  $\begin{array}{l}r\left(0\right)=\frac{{\left(0\right)}^2-9}{{\left(0\right)}^2}\\ r\left(0\right)=\infty \end{array}$

Since, the $y$ -intercept of the rational function is occurring when $y\to \infty$ .

So, there is no $y$ -intercept of the rational function.

Hence,

The $x$ -interceptare $3\text{ and }-3$ and there isno $y$ -interceptof the given rational function.