To determine the domain of the composition of functions [tex]\((g \circ f)(x)\)[/tex], let's analyze the given information and the restrictions for both functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
1. The domain of [tex]\(f(x)\)[/tex] comprises all real numbers except 7. This means [tex]\(f(x)\)[/tex] is defined for all real [tex]\(x\)[/tex] except when [tex]\(x = 7\)[/tex].
2. The domain of [tex]\(g(x)\)[/tex] comprises all real numbers except -3. This means [tex]\(g(x)\)[/tex] is defined for all real [tex]\(x\)[/tex] except when its input (which is [tex]\(f(x)\)[/tex] in this case) is -3.
For the composition [tex]\((g \circ f)(x)\)[/tex], which means [tex]\(g(f(x))\)[/tex], to be defined, both [tex]\(f(x)\)[/tex] and [tex]\(g(f(x))\)[/tex] must be defined.
Let's break it down step-by-step:
1. [tex]\(f(x)\)[/tex] must be defined:
- [tex]\(f(x)\)[/tex] is defined for all [tex]\(x \neq 7\)[/tex]. Hence, [tex]\(x = 7\)[/tex] should be excluded from the domain of [tex]\((g \circ f)(x)\)[/tex].
2. [tex]\(g(f(x))\)[/tex] must be defined:
- [tex]\(g(x)\)[/tex] is defined for all inputs except -3. Therefore, [tex]\(f(x)\)[/tex] must not be -3 for [tex]\(g(f(x))\)[/tex] to be defined. This means that any [tex]\(x\)[/tex] that makes [tex]\(f(x) = -3\)[/tex] should be excluded from the domain of [tex]\((g \circ f)(x)\)[/tex].
Combining these two points, the domain of [tex]\((g \circ f)(x)\)[/tex] will exclude:
- [tex]\(x = 7\)[/tex] (since [tex]\(f(x)\)[/tex] is not defined at [tex]\(x = 7\)[/tex])
- Any [tex]\(x\)[/tex] such that [tex]\(f(x) = -3\)[/tex] (since [tex]\(g(x)\)[/tex] is not defined when its argument is -3)
Upon evaluating the given multiple-choice options, we see that the third choice accurately reflects these conditions:
- All real values except [tex]\(x = 7\)[/tex] and the [tex]\(x\)[/tex] for which [tex]\(f(x) = 7\)[/tex].
The correct answer is:
All real values except [tex]\(x = 7\)[/tex] and the [tex]\(x\)[/tex] for which [tex]\(f(x) = -3\)[/tex].
Thus, the choice that fits this description is:
3. All real values except [tex]\(x = 7\)[/tex] and the [tex]\(x\)[/tex] for which [tex]\(f(x) = -3\)[/tex].
