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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]\( x = -3 \)[/tex] and the [tex]\( x \)[/tex] for which [tex]\( f(x) = 7 \)[/tex]
B. All real values except [tex]\( x = -3 \)[/tex] and the [tex]\( x \)[/tex] for which [tex]\( f(x) = -3 \)[/tex]
C. All real values except [tex]\( x = 7 \)[/tex] and the [tex]\( x \)[/tex] for which [tex]\( f(x) = 7 \)[/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
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].
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