To determine the domain of the composite function [tex]\((g \circ f)(x)\)[/tex], we need to consider the domains of both [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex].
1. The domain of [tex]\(f(x)\)[/tex] is all real values except 7. This means [tex]\(f(x)\)[/tex] is not defined at [tex]\(x = 7\)[/tex].
2. The domain of [tex]\(g(x)\)[/tex] is all real values except -3. This implies [tex]\(g(x)\)[/tex] is not defined at [tex]\(x = -3\)[/tex].
For the composite function [tex]\( (g \circ f)(x) = g(f(x)) \)[/tex] to be defined at some [tex]\( x \)[/tex]:
- [tex]\( f(x) \)[/tex] must be defined at [tex]\( x \)[/tex]. So, [tex]\( x \)[/tex] cannot be 7.
- [tex]\( g(f(x)) \)[/tex] must be defined, which means [tex]\( f(x) \)[/tex] cannot be -3 because [tex]\( g(x) \)[/tex] is not defined at -3.
Hence, the domain of [tex]\( (g \circ f)(x) \)[/tex] is:
- All real values except [tex]\( x \)[/tex] where [tex]\( x = 7 \)[/tex].
- All real values except [tex]\( x \)[/tex] such that [tex]\( f(x) = -3 \)[/tex].
Taking all these points into account, the correct description of the domain of [tex]\((g \circ f)(x)\)[/tex] is:
All real values except [tex]\( x \neq 7 \)[/tex] and the [tex]\( x \)[/tex] for which [tex]\( f(x) \neq -3 \)[/tex].
Therefore, 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].
