To determine the domain of the composite function [tex]\((g \circ f)(x)\)[/tex], we must consider the individual domains of [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex], and how they interact.
1. Domain of [tex]\(f(x)\)[/tex]:
- Given: The domain of [tex]\(f(x)\)[/tex] is all real values except 7.
- This means [tex]\(f(x)\)[/tex] is undefined when [tex]\(x = 7\)[/tex].
2. Domain of [tex]\(g(x)\)[/tex]:
- Given: The domain of [tex]\(g(x)\)[/tex] is all real values except -3.
- This means [tex]\(g(y)\)[/tex] is undefined when [tex]\(y = -3\)[/tex].
Now, to find the domain of [tex]\((g \circ f)(x)\)[/tex], we need to ensure two things:
1. [tex]\(f(x)\)[/tex] must be defined.
2. [tex]\(g(f(x))\)[/tex] must be defined.
This means:
- The input [tex]\(x\)[/tex] must not make [tex]\(f(x)\)[/tex] undefined. Thus, [tex]\(x \neq 7\)[/tex] because [tex]\(f(x)\)[/tex] is not defined at [tex]\(x = 7\)[/tex].
- [tex]\(f(x)\)[/tex] must not take a value that makes [tex]\(g(y)\)[/tex] undefined. Therefore, [tex]\(f(x) \neq -3\)[/tex] because [tex]\(g(x)\)[/tex] is not defined at [tex]\(x = -3\)[/tex].
So, the domain of [tex]\((g \circ f)(x)\)[/tex] is all real values except:
- [tex]\(x = 7\)[/tex] and
- The values of [tex]\(x\)[/tex] for which [tex]\(f(x) = -3\)[/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].
This corresponds to the option:
- "all real values except [tex]\(x \neq 7\)[/tex] and the [tex]\(x\)[/tex] for which [tex]\(f(x) \neq -3\)[/tex]"
