Given the functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex], we know the following constraints:
1. The domain of [tex]\( f(x) \)[/tex] is all real values except 7. This means [tex]\( f(x) \)[/tex] is undefined at [tex]\( x = 7 \)[/tex].
2. The domain of [tex]\( g(x) \)[/tex] is all real values except -3. This means [tex]\( g(x) \)[/tex] is undefined at [tex]\( x = -3 \)[/tex].
We are asked to find the domain of the composition function [tex]\( (g \circ f)(x) \)[/tex], which means [tex]\( g(f(x)) \)[/tex].
### Step-by-Step Solution:
1. Understand the Composition:
In the composition [tex]\( (g \circ f)(x) \)[/tex], [tex]\( f(x) \)[/tex] is evaluated first and then [tex]\( g \)[/tex] is applied to the result of [tex]\( f(x) \)[/tex].
2. Initial Domain Restriction from [tex]\( f(x) \)[/tex]:
Since [tex]\( f(x) \)[/tex] is not defined for [tex]\( x = 7 \)[/tex], we must exclude [tex]\( x = 7 \)[/tex] from the domain of [tex]\( (g \circ f)(x) \)[/tex].
3. Domain Restriction from [tex]\( g(x) \)[/tex]:
For [tex]\( g(f(x)) \)[/tex] to be defined, [tex]\( f(x) \)[/tex] must be within the domain of [tex]\( g \)[/tex]. This means the output of [tex]\( f(x) \)[/tex] must not be -3 because [tex]\( g \)[/tex] is undefined at -3.
Therefore, we need to ensure [tex]\( f(x) \neq -3 \)[/tex].
4. Combining the Restrictions:
- We exclude [tex]\( x = 7 \)[/tex] from the domain because [tex]\( f(x) \)[/tex] is undefined at [tex]\( x = 7 \)[/tex].
- We also need to ensure the result of [tex]\( f(x) \)[/tex] is not -3, so [tex]\( x \)[/tex] must not be such that [tex]\( f(x) = -3 \)[/tex].
Thus, the domain of [tex]\( (g \circ f)(x) \)[/tex] is all real values except:
- [tex]\( x = 7 \)[/tex], and
- [tex]\( x \)[/tex] for which [tex]\( f(x) = -3 \)[/tex].
### Conclusion:
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].
The answer is:
All real values except [tex]\( x \neq 7 \)[/tex] and the [tex]\( x \)[/tex] for which [tex]\( f(x) \neq -3 \)[/tex].
