Question

find the domain of the function.
$f(x)=\frac{1}{x^2 + 25}-\frac{4}{x^2 - 9}$
what is the domain of f?
(type your answer in interval notation.)

Explanation:

Step1: Identify undefined denominator values

For rational functions, denominators cannot equal 0. First, check $x^2 + 25 = 0$:
$x^2 = -25$ has no real solutions.
Now check $x^2 - 9 = 0$:
$x^2 = 9$
$x = \pm 3$

Step2: Exclude undefined x-values

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

Step3: Write in interval notation

We split the real line into intervals excluding $-3$ and $3$:
$(-\infty, -3) \cup (-3, 3) \cup (3, \infty)$

Answer:

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