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) \neq-3[/tex]
Answer (2)
The domain of f ( x ) is all real numbers except 7, so x e q 7 .
The domain of g ( x ) is all real numbers except -3, so f ( x ) e q − 3 .
The domain of ( g c i rc f ) ( x ) is all real numbers except x e q 7 and the x for which f ( x ) = − 3 .
Therefore, the answer is all real values except x e q 7 and the x for which f ( x ) e q − 3 . all real values except x = 7 and the x for which f ( x ) = − 3
Explanation
Understanding the Domain of Composite Functions 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 )) . This means we need to consider two restrictions:
Since x is an input to f , x cannot be 7 (because 7 is not in the domain of f ).
Since f ( x ) is an input to g , f ( x ) cannot be -3 (because -3 is not in the domain of g ).
Therefore, the domain of ( g ∘ f ) ( x ) is all real numbers except x = 7 and the x values for which f ( x ) = − 3 .
Analyzing Domain Restrictions The domain of f ( x ) is all real numbers except 7, which means x = 7 .
The domain of g ( x ) is all real numbers except -3, which means g ( x ) is not defined when its input is -3. Thus, we require f ( x ) = − 3 .
Therefore, the domain of ( g ∘ f ) ( x ) is all real numbers except x = 7 and the x values for which f ( x ) = − 3 .
Determining the Domain of (g \circ f)(x) The domain of ( g ∘ f ) ( x ) is all real numbers except x = 7 and the x values for which f ( x ) = − 3 . This corresponds to the statement '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 given x amount of raw materials, and g ( y ) represents the profit the company makes given y number of products. If the factory cannot accept 7 units of raw materials and the company cannot handle f ( x ) = − 3 products (perhaps due to a loss), then the composite function g ( f ( x )) helps determine the overall domain of raw materials the factory can accept to make a profit.
The domain of the composite function ( g ∘ f ) ( x ) includes all real numbers except for values where x = 7 and where f ( x ) = − 3 . Therefore, the correct answer is option A.
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