To determine the domain of the composition of two functions [tex]\( (g \circ f)(x) \)[/tex], we need to consider when each function is defined.
1. Domain of [tex]\( f(x) \)[/tex]:
- [tex]\( f(x) \)[/tex] is defined for all real values except [tex]\( x = 7 \)[/tex].
- So, the domain of [tex]\( f(x) \)[/tex] is [tex]\( \mathbb{R} \setminus \{ 7 \} \)[/tex].
2. Domain of [tex]\( g(x) \)[/tex]:
- [tex]\( g(x) \)[/tex] is defined for all real values except [tex]\( x = -3 \)[/tex].
- So, the domain of [tex]\( g(x) \)[/tex] is [tex]\( \mathbb{R} \setminus \{ -3 \} \)[/tex].
3. Domain of the composition [tex]\( (g \circ f)(x) = g(f(x)) \)[/tex]:
- For [tex]\( g(f(x)) \)[/tex] to be defined, two conditions must be met:
- [tex]\( f(x) \)[/tex] must be defined.
- [tex]\( g(y) \)[/tex] must be defined where [tex]\( y = f(x) \)[/tex].
- First, [tex]\( f(x) \)[/tex] must be defined, so [tex]\( x \neq 7 \)[/tex].
- Second, [tex]\( g(f(x)) \)[/tex] must be defined. Since [tex]\( g(y) \)[/tex] is not defined for [tex]\( y = -3 \)[/tex], we need to ensure that [tex]\( f(x) \neq -3 \)[/tex].
Therefore, the domain of [tex]\( (g \circ f)(x) \)[/tex] consists of all real numbers [tex]\( x \)[/tex] except where [tex]\( x = 7 \)[/tex] and where [tex]\( f(x) = -3 \)[/tex].
So, the correct description of the domain of [tex]\( (g \circ f)(x) \)[/tex] is:
All real values except [tex]\( x \neq 7 \)[/tex] and [tex]\( x \)[/tex] for which [tex]\( f(x) \neq -3 \)[/tex].
