To determine the values of [tex]\( x \)[/tex] that are not in the domain of the function [tex]\( g(x) = \frac{x-3}{x^2 - 16} \)[/tex], we need to identify where the function is undefined. The function [tex]\( g(x) \)[/tex] is undefined wherever its denominator is zero since division by zero is not defined in mathematics.
Let's start with the denominator of the function:
[tex]\[
x^2 - 16
\][/tex]
We need to find the values of [tex]\( x \)[/tex] that make the denominator zero. Set the denominator equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[
x^2 - 16 = 0
\][/tex]
To solve this equation, notice that it is a difference of squares, which can be factored as follows:
[tex]\[
x^2 - 16 = (x - 4)(x + 4) = 0
\][/tex]
Next, set each factor equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[
x - 4 = 0 \quad \text{or} \quad x + 4 = 0
\][/tex]
Solve these simple equations:
[tex]\[
x = 4 \quad \text{or} \quad x = -4
\][/tex]
Therefore, the function [tex]\( g(x) \)[/tex] is undefined at [tex]\( x = 4 \)[/tex] and [tex]\( x = -4 \)[/tex]. These are the values that are not in the domain of the function.
In conclusion, the values of [tex]\( x \)[/tex] that are not in the domain of [tex]\( g \)[/tex] are [tex]\( \boxed{-4, 4} \)[/tex].
