To determine the domain of the composite function [tex]\((g \circ f)(x)\)[/tex], we need to consider the domains of both functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex].
Here's a detailed, step-by-step solution:
1. Domain of [tex]\(f(x)\)[/tex]: The domain of [tex]\(f(x)\)[/tex] is all real values except 7. Therefore, we can write:
[tex]\[
D_f = \{ x \in \mathbb{R} \mid x \neq 7 \}
\][/tex]
2. Domain of [tex]\(g(x)\)[/tex]: The domain of [tex]\(g(x)\)[/tex] is all real values except -3. Therefore, we can write:
[tex]\[
D_g = \{ y \in \mathbb{R} \mid y \neq -3 \}
\][/tex]
3. Composite function [tex]\((g \circ f)(x)\)[/tex]: For the composite function [tex]\((g \circ f)(x)\)[/tex] to be defined at a specific value [tex]\(x\)[/tex], two conditions must be met:
- [tex]\(x\)[/tex] must be in the domain of [tex]\(f\)[/tex]. From step 1, [tex]\(x \neq 7\)[/tex].
- [tex]\(f(x)\)[/tex] must be in the domain of [tex]\(g\)[/tex]. From step 2, [tex]\(f(x) \neq -3\)[/tex].
4. Combining the Conditions:
- [tex]\(x\)[/tex] must not be 7 because [tex]\(x = 7\)[/tex] is not in the domain of [tex]\(f\)[/tex].
- [tex]\(f(x) \neq -3\)[/tex] because [tex]\(f(x) = -3\)[/tex] is not in the domain of [tex]\(g\)[/tex].
Putting these conditions together, the domain of the composite function [tex]\((g \circ f)(x)\)[/tex] can be described as the set of all real values [tex]\(x\)[/tex] such that:
- [tex]\(x \neq 7\)[/tex]
- The value [tex]\(f(x) \neq -3\)[/tex]
Therefore, the domain of [tex]\((g \circ f)(x)\)[/tex] is:
[tex]\[
\text{All real values except } x \neq 7 \text{ and the } x \text{ for which } f(x) \neq -3
\][/tex]
Hence, the correct option is:
- All real values except [tex]\( x \neq 7 \)[/tex] and the [tex]\( x \)[/tex] for which [tex]\( f(x) \neq -3 \)[/tex].
