To find the domain of the function [tex]\( g(x) = \frac{x+9}{x^2 - 4} \)[/tex], we need to identify the values of [tex]\( x \)[/tex] for which the function is defined. Specifically, the function is undefined whenever the denominator is zero, as division by zero is not allowed.
1. Identify the denominator of the function:
[tex]\[
x^2 - 4
\][/tex]
2. Set the denominator equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[
x^2 - 4 = 0
\][/tex]
[tex]\[
(x - 2)(x + 2) = 0
\][/tex]
[tex]\[
x - 2 = 0 \quad \text{or} \quad x + 2 = 0
\][/tex]
[tex]\[
x = 2 \quad \text{or} \quad x = -2
\][/tex]
So, [tex]\( x = 2 \)[/tex] and [tex]\( x = -2 \)[/tex] are the values that make the denominator zero.
3. Determine the domain by excluding these values:
The domain of [tex]\( g(x) \)[/tex] is all real numbers except [tex]\( x = -2 \)[/tex] and [tex]\( x = 2 \)[/tex].
4. Express the domain in interval notation:
The real line is divided into three intervals around the points where the function is undefined:
[tex]\[
(-\infty, -2), \quad (-2, 2), \quad \text{and} \quad (2, \infty)
\][/tex]
Therefore, the domain of [tex]\( g(x) \)[/tex] is:
[tex]\[
(-\infty, -2) \cup (-2, 2) \cup (2, \infty)
\][/tex]
Thus, the domain of the function [tex]\( g(x) = \frac{x+9}{x^2-4} \)[/tex] is [tex]\(\boxed{(-\infty, -2) \cup (-2, 2) \cup (2, \infty)}\)[/tex].
