Question

find the domain of the function. $f(x)=\frac{6}{x^{2}+16}+\frac{4}{x^{2}-9}$ what is the domain of f? (type your answer in interval notation.)

Explanation:

Step1: Identify undefined denominator values

For $\frac{6}{x^2+16}$, $x^2+16=0$ has no real solutions since $x^2\geq0$, so $x^2+16\geq16>0$.
For $\frac{4}{x^2-9}$, set $x^2-9=0$.
$x^2-9=0 \implies (x-3)(x+3)=0 \implies x=3 \text{ or } x=-3$

Step2: Exclude undefined x-values

The function is undefined only at $x=3$ and $x=-3$. All other real numbers are allowed.

Step3: Write domain in interval notation

Combine the intervals of allowed x-values: $(-\infty,-3) \cup (-3,3) \cup (3,\infty)$

Answer:

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