To determine the domain of the composite function [tex]\((f \circ g)(x)\)[/tex], which is [tex]\(f(g(x))\)[/tex], we need to understand the domains of both functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex] and how they affect each other.
1. Understand the domain of [tex]\(g(x)\)[/tex]:
- The function [tex]\(g(x) = \frac{1}{x-13}\)[/tex] is defined for all real numbers except where the denominator is zero.
- The denominator [tex]\(x-13\)[/tex] is zero when [tex]\(x = 13\)[/tex].
- Therefore, the domain of [tex]\(g(x)\)[/tex] is all real numbers except [tex]\(x = 13\)[/tex].
2. Identify the domain of [tex]\(f(g(x))\)[/tex]:
- The function [tex]\(f(x) = x + 7\)[/tex] is a simple linear function, which is defined for all real numbers.
- Since [tex]\(f(x)\)[/tex] has no additional restrictions, the only restriction on [tex]\(f(g(x))\)[/tex] comes from the restriction on [tex]\(g(x)\)[/tex].
- Hence, [tex]\(f(g(x))\)[/tex] will be defined wherever [tex]\(g(x)\)[/tex] is defined.
3. Conclusion:
- Since [tex]\(g(x)\)[/tex] is defined for all [tex]\(x\)[/tex] except [tex]\(x = 13\)[/tex], the composite function [tex]\((f \circ g)(x)\)[/tex] will also be defined for all [tex]\(x\)[/tex] except [tex]\(x = 13\)[/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].
