Chapter 8.3, Problem 8E

Solution

To identify: The asymptotes of the graph of the function $f\left(x\right)=\frac{x^2-9}{x+3}$ .

No horizontal or vertical asymptotes.

Given information:

The given function is $f\left(x\right)=\frac{x^2-9}{x+3}$ .

Explanation:

Consider the given function.

If degree of numerator is greater than the degree of denominator then the function has no horizontal asymptotes.

Since the given function’s numerator degree is greater than denominator’s degree so the function has no horizontal asymptotes.

Simplify the given function:

  $\begin{array}{c}f\left(x\right)=\frac{x^2-9}{x+3}\\ =\frac{\left(x-3\right)\left(x+3\right)}{\left(x+3\right)}\\ =x-3\end{array}$

Numerator and denominator have common factor which is $x+3$ thus, the function has a hole at $x=-3$