To determine the domain of [tex]\((f \circ g)(x)\)[/tex], we need to understand the individual domains of the functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex] and then find where the composition is defined.
1. First, consider the function [tex]\(g(x) = \frac{1}{x - 13}\)[/tex].
- The domain of [tex]\(g(x)\)[/tex] is all real numbers except where the denominator is zero.
- So, [tex]\(x - 13 \neq 0\)[/tex] which means [tex]\(x \neq 13\)[/tex].
- Therefore, the domain of [tex]\(g(x)\)[/tex] is [tex]\(\{x \mid x \neq 13\}\)[/tex].
2. Next, consider the function [tex]\(f(x) = x + 7\)[/tex].
- The function [tex]\(f(x)\)[/tex] is defined for all real numbers.
- Therefore, the domain of [tex]\(f(x)\)[/tex] is all real numbers, [tex]\(\mathbb{R}\)[/tex].
3. Now, we consider the composition [tex]\((f \circ g)(x)\)[/tex], which is [tex]\(f(g(x))\)[/tex].
- The composition [tex]\((f \circ g)(x)\)[/tex] is defined wherever [tex]\(g(x)\)[/tex] is defined and [tex]\(g(x)\)[/tex] produces a result within the domain of [tex]\(f\)[/tex].
Since the domain of [tex]\(g(x)\)[/tex] is [tex]\(\{x \mid x \neq 13\}\)[/tex], and [tex]\(f\)[/tex] is defined for all real numbers, the only restriction for the composition [tex]\((f \circ g)(x)\)[/tex] comes from the restriction on [tex]\(g(x)\)[/tex].
Thus, the domain of [tex]\((f \circ g)(x)\)[/tex] is [tex]\(\{x \mid x \neq 13\}\)[/tex].
The correct answer is:
[tex]\(\{x \mid x \neq 13\}\)[/tex]
