To find the domain of the rational function [tex]\( h(x) = \frac{x+8}{x^2 - 64} \)[/tex]:
1. Identify Restrictions: The domain of a rational function is all real numbers except where the denominator is zero. So, we start by setting the denominator equal to zero and solving for [tex]\( x \)[/tex]:
[tex]\[
x^2 - 64 = 0
\][/tex]
2. Solve for [tex]\( x \)[/tex]:
[tex]\[
x^2 - 64 = (x - 8)(x + 8) = 0
\][/tex]
Setting each factor equal to zero:
[tex]\[
x - 8 = 0 \quad \Rightarrow \quad x = 8
\][/tex]
[tex]\[
x + 8 = 0 \quad \Rightarrow \quad x = -8
\][/tex]
3. Determine Domain:
Since [tex]\( x = 8 \)[/tex] and [tex]\( x = -8 \)[/tex] make the denominator zero, these values are excluded from the domain.
The domain excludes [tex]\( x = 8 \)[/tex] and [tex]\( x = -8 \)[/tex], so the domain in interval notation is all real numbers except [tex]\( -8 \)[/tex] and [tex]\( 8 \)[/tex].
We can write the domain as:
[tex]\[
(-\infty, -8) \cup (-8, 8) \cup (8, \infty)
\][/tex]
Therefore, the domain of [tex]\( h(x) \)[/tex] is restricted to:
[tex]\[
\boxed{(-\infty, -8) \cup (-8, 8) \cup (8, \infty)}
\][/tex]
So the correct choice is:
A. The domain of [tex]\( h(x) \)[/tex] is restricted to [tex]\(\boxed{(-\infty, -8) \cup (-8, 8) \cup (8, \infty)} \)[/tex].
