To determine the domain of [tex]\((g \circ f)(x)\)[/tex], we need to recall the general rule for composing functions: the domain of [tex]\((g \circ f)(x)\)[/tex] is the set of all [tex]\(x\)[/tex] in the domain of [tex]\(f\)[/tex] such that [tex]\(f(x)\)[/tex] lies in the domain of [tex]\(g\)[/tex].
Given:
- The domain of [tex]\(f(x)\)[/tex] is all real values except 7.
- The domain of [tex]\(g(x)\)[/tex] is all real values except -3.
We need to exclude any values of [tex]\(x\)[/tex] that make [tex]\(f(x) = 7\)[/tex], as [tex]\(x = 7\)[/tex] is not allowed in the domain of [tex]\(f\)[/tex]. Additionally, we need to exclude any [tex]\(x\)[/tex] such that [tex]\(f(x) = -3\)[/tex], because -3 is not allowed in the domain of [tex]\(g\)[/tex].
Therefore, the domain of [tex]\((g \circ f)(x)\)[/tex] includes all real values except:
- [tex]\(x = 7\)[/tex]
- The [tex]\(x\)[/tex] values for which [tex]\(f(x) = -3\)[/tex]
From this, we see that the correct description matches with the solution we obtained:
All real values except [tex]\( x = 7 \)[/tex] and the [tex]\(x\)[/tex] for which [tex]\( f(x) = -3 \)[/tex].
Thus, the correct option is:
all real values except [tex]\(x = 7\)[/tex] and the [tex]\(x\)[/tex] for which [tex]\(f(x) = -3\)[/tex].
