To find the domain of the composition function [tex]\((f \circ g)(x)\)[/tex] where [tex]\(f(x) = x + 7\)[/tex] and [tex]\(g(x) = \frac{1}{x - 13}\)[/tex], we need to carefully look at the individual functions and their domains, and then derive the domain for the composition. Here are the step-by-step details:
1. Domain of [tex]\(g(x)\)[/tex]:
[tex]\(g(x) = \frac{1}{x - 13}\)[/tex] is a fraction, and its denominator cannot be zero. Therefore, [tex]\(x - 13 \neq 0\)[/tex].
Thus, [tex]\( x \neq 13\)[/tex].
So, the domain of [tex]\(g(x)\)[/tex] is all real numbers except [tex]\(x = 13\)[/tex]:
[tex]\[
\text{Domain of } g(x): \{x \mid x \neq 13\}
\][/tex]
2. Expression for [tex]\((f \circ g)(x)\)[/tex]:
The composition [tex]\((f \circ g)(x)\)[/tex] means applying [tex]\(g(x)\)[/tex] first and then applying [tex]\(f(x)\)[/tex] to the result:
[tex]\[
(f \circ g)(x) = f(g(x)) = f\left(\frac{1}{x - 13}\right)
\][/tex]
3. Simplify [tex]\(f(g(x))\)[/tex]:
Now substitute [tex]\(g(x)\)[/tex] into [tex]\(f(x)\)[/tex]:
[tex]\[
f\left(\frac{1}{x - 13}\right) = \frac{1}{x - 13} + 7
\][/tex]
This new function must also have no restrictions imposed by [tex]\(f(x)\)[/tex].
4. Domain of [tex]\(f(g(x))\)[/tex]:
Since [tex]\(\frac{1}{x - 13}\)[/tex] must be defined first, [tex]\(x \neq 13\)[/tex], we have to consider if there are any additional conditions imposed by the composition resulting in a restriction on the domain.
In this particular case, there is no numerator restriction added by the function [tex]\(f\)[/tex]. Thus, as long as [tex]\(g(x)\)[/tex] is defined, [tex]\((f \circ g)(x)\)[/tex] will also be defined.
So, the domain of [tex]\((f \circ g)(x) = f(g(x))\)[/tex] takes into account the restriction from [tex]\(g(x)\)[/tex]:
[tex]\[
\text{Domain of } (f \circ g)(x): \{x \mid x \neq 13\}
\][/tex]
Therefore, the domain of [tex]\((f \circ g)(x)\)[/tex] is:
[tex]\[
\boxed{\{x \mid x \neq 13\}}
\][/tex]
