To find the values of [tex]\( x \)[/tex] that are not in the domain of the function [tex]\( h(x) = \frac{x^2 - 9x + 18}{x^2 - 81} \)[/tex], we need to identify where the function has undefined values. A function is undefined whenever the denominator is equal to zero.
First, take a look at the denominator of the function:
[tex]\[
x^2 - 81
\][/tex]
We need to find the values of [tex]\( x \)[/tex] for which the denominator is zero:
[tex]\[
x^2 - 81 = 0
\][/tex]
This equation can be factored using the difference of squares:
[tex]\[
(x - 9)(x + 9) = 0
\][/tex]
Setting each factor equal to zero gives:
[tex]\[
x - 9 = 0 \quad \text{or} \quad x + 9 = 0
\][/tex]
Solving these equations, we find:
[tex]\[
x = 9 \quad \text{or} \quad x = -9
\][/tex]
Therefore, the values of [tex]\( x \)[/tex] that make the denominator zero, and thus are not in the domain of [tex]\( h(x) \)[/tex], are [tex]\( x = 9 \)[/tex] and [tex]\( x = -9 \)[/tex].
So, the values of [tex]\( x \)[/tex] that are not in the domain of the function [tex]\( h \)[/tex] are:
[tex]\[
-9, 9
\][/tex]
