Question

sketch the graph of the rational function. note that the function is not in lowest terms. find the domain first. f(x)=\frac{x + 3}{x^{2}-9} what is the domain of the rational function? (type your answer in interval notation.)

Explanation:

Step1: Recall domain - non - zero denominator

The denominator of the rational function $f(x)=\frac{x + 3}{x^{2}-9}$ cannot be zero. We know that $x^{2}-9=(x + 3)(x - 3)$.

Step2: Solve for values of x that make denominator zero

Set $(x + 3)(x - 3)=0$. Using the zero - product property, if $ab = 0$, then $a=0$ or $b = 0$. So $x+3=0$ gives $x=-3$ and $x - 3=0$ gives $x = 3$.

Step3: Write domain in interval notation

The domain is all real numbers except $x=-3$ and $x = 3$. In interval notation, it is $(-\infty,-3)\cup(-3,3)\cup(3,\infty)$.

Answer:

$(-\infty,-3)\cup(-3,3)\cup(3,\infty)$