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 \neq 7[/tex] and the [tex]x[/tex] for which [tex]f(x) \neq 7[/tex]
D. all real values except [tex]x \neq 7[/tex] and the [tex]x[/tex] for which [tex]f(x)=-3[/tex]
Answer (2)
The domain of f ( x ) excludes x = 7 .
The domain of g ( x ) excludes x = − 3 , so we must exclude x such that f ( x ) = − 3 .
Therefore, the domain of $(g
\circ f)(x) e x c l u d es x = 7 an d t h e x f or w hi c h f(x) = -3$.
The final answer is all real values except x e q 7 and the x for which f ( x ) = − 3 .
Explanation
Understanding the Problem We are given that the domain of f ( x ) is all real numbers except 7, and the domain of g ( x ) is all real numbers except -3. We want to find the domain of the composite function ( g ∘ f ) ( x ) = g ( f ( x )) .
Considering the Domain of f(x) The domain of g ( f ( x )) consists of all x in the domain of f such that f ( x ) is in the domain of g . Since the domain of f ( x ) is all real numbers except 7, we must exclude x = 7 from the domain of ( g ∘ f ) ( x ) .
Considering the Domain of g(x) Since the domain of g ( x ) is all real numbers except -3, we must exclude all x such that f ( x ) = − 3 from the domain of ( g ∘ f ) ( x ) . In other words, we need to find the values of x for which f ( x ) = − 3 and exclude them from the domain.
Determining the Domain of the Composite Function Therefore, the domain of ( g ∘ f ) ( x ) is all real numbers except x = 7 and the x for which f ( x ) = − 3 .
Final Answer The correct answer is 'all real values except x = 7 and the x for which f ( x ) = − 3 '.
Examples
Consider a scenario where f ( x ) represents the number of products a factory can produce based on the number of employees x , and g ( x ) represents the profit made from selling x products. If the factory cannot function with exactly 7 employees (domain of f ( x ) excludes 7) and the profit calculation has an issue when the number of products is -3 (domain of g ( x ) excludes -3), then the composite function g ( f ( x )) represents the profit based on the number of employees. The domain of g ( f ( x )) would exclude 7 employees and any number of employees that would result in -3 products, ensuring the composite function is valid and meaningful. This helps in understanding the constraints and limitations when combining different processes or functions in real-world applications.
The domain of the composite function ( g ∘ f ) ( x ) excludes all values of x where f ( x ) = − 3 and also excludes x = 7 . Therefore, the answer is all real values except x = 7 and values for which f ( x ) = − 3 . The correct option is D.
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