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The domain of [tex]f(x)[/tex] is the set of all real values except 7, and the domain of [tex]g(x)[/tex] is the set of all real values except -3. Which of the following describes the domain of [tex](g \circ f)(x)[/tex]?

A. all real values except [tex]x \neq -3[/tex] and the [tex]x[/tex] for which [tex]f(x) \neq 7[/tex]
B. all real values except [tex]x \neq -3[/tex] and the [tex]x[/tex] for which [tex]f(x) \neq -3[/tex]
C. all real values except [tex]x = 7[/tex] and the [tex]x[/tex] for which [tex]f(x) \approx 7[/tex]
D. all real values except [tex]x \neq 7[/tex] and the [tex]x[/tex] for which [tex]f(x) \neq -3[/tex]

Sagot :

GinnyAnswer
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].
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