To determine the domain of the composition of functions [tex]\((f \circ g)(x)\)[/tex], we first need to understand the individual domains of [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex], and then see how these domains interact in the composition.
Given the functions:
[tex]\[ f(x) = x + 7 \][/tex]
[tex]\[ g(x) = \frac{1}{x - 13} \][/tex]
Let's compose the functions:
[tex]\[ (f \circ g)(x) = f(g(x)) \][/tex]
Now substitute [tex]\(g(x)\)[/tex] into [tex]\(f(x)\)[/tex]:
[tex]\[ f(g(x)) = f\left(\frac{1}{x - 13}\right) \][/tex]
[tex]\[ f\left(\frac{1}{x - 13}\right) = \left( \frac{1}{x - 13} \right) + 7 \][/tex]
To find the domain of [tex]\( (f \circ g)(x) \)[/tex], we need to determine where this expression is defined. The key is to remember that [tex]\(g(x)\)[/tex] involves a division, and division by zero is undefined. So, we need [tex]\(g(x)\)[/tex] to be defined.
For [tex]\(g(x) = \frac{1}{x - 13} \)[/tex] to be defined, the denominator cannot be zero. Hence:
[tex]\[ x - 13 \neq 0 \][/tex]
[tex]\[ x \neq 13 \][/tex]
Thus, the domain of [tex]\(g(x)\)[/tex] is all real numbers except [tex]\(13\)[/tex].
Since [tex]\( (f \circ g)(x) = f(g(x)) \)[/tex] is defined whenever [tex]\(g(x)\)[/tex] is defined, the domain of the composite function [tex]\( (f \circ g)(x) \)[/tex] is all real numbers except where [tex]\( g(x) \)[/tex] is undefined. Therefore, the domain of [tex]\( (f \circ g)(x) \)[/tex] is:
[tex]\[ \{x \mid x \neq 13\} \][/tex]
Thus, the correct answer is:
[tex]\[ \{x \mid x \neq 13\} \][/tex]
