Question

find the domain of the function. f(x)=\frac{6}{x^{2}+25}-\frac{9}{x^{2}-9} what is the domain of f? (type your answer in interval notation.)

Explanation:

Step1: Identify values that make denominator 0

Set $x^{2}+25 = 0$ and $x^{2}-9=0$. For $x^{2}+25 = 0$, we have $x^{2}=- 25$, no real - valued solutions. For $x^{2}-9 = 0$, factor to $(x + 3)(x - 3)=0$, so $x=-3$ or $x = 3$.

Step2: Determine the domain

The domain of a rational function is all real numbers except the values that make the denominator 0. Since the only real - valued values that make the denominator 0 are $x=-3$ and $x = 3$, the domain is $(-\infty,-3)\cup(-3,3)\cup(3,\infty)$.

Answer:

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