To determine the domain of the composition of two functions [tex]\((g \circ f)(x)\)[/tex], we need to consider the domains of both [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex], along with how they interact in the composition.
Given:
- The domain of [tex]\(f(x)\)[/tex] is all real values except 7. This means [tex]\(f(x)\)[/tex] is undefined at [tex]\(x = 7\)[/tex].
- The domain of [tex]\(g(x)\)[/tex] is all real values except -3. This means [tex]\(g(x)\)[/tex] is undefined at [tex]\(x = -3\)[/tex].
For [tex]\((g \circ f)(x)\)[/tex] (i.e., [tex]\(g(f(x))\)[/tex]) to be defined:
1. [tex]\(f(x)\)[/tex] must be defined, which excludes [tex]\(x = 7\)[/tex].
2. Additionally, [tex]\(f(x)\)[/tex] must produce a value that is within the domain of [tex]\(g(x)\)[/tex]. Since [tex]\(g(x)\)[/tex] is undefined at [tex]\(x = -3\)[/tex], [tex]\(f(x)\)[/tex] must not be -3.
Therefore, the domain of [tex]\((g \circ f)(x)\)[/tex] is all real values except:
- [tex]\(x = 7\)[/tex] (because [tex]\(f(x)\)[/tex] is not defined there), and
- The values of [tex]\(x\)[/tex] for which [tex]\(f(x) = -3\)[/tex] (because [tex]\(g(x)\)[/tex] is not defined for [tex]\(f(x) = -3\)[/tex]).
Among the given options:
- "All real values except [tex]\(x \neq 7\)[/tex] and the [tex]\(x\)[/tex] for which [tex]\(f(x) \neq -3\)[/tex]" correctly matches our conclusion.
Thus, the correct answer is:
All real values except [tex]\(x \neq 7\)[/tex] and the [tex]\(x\)[/tex] for which [tex]\(f(x) \neq -3\)[/tex].
