The domain of $f(x)$ is the set of all real values except 7, and the domain of $g(x)$ is the set of all real values except -3. Which of the following describes the domain of $(g \circ f)(x)$?
A. all real values except $x \neq -3$ and the $x$ for which $f(x) \neq 7$
B. all real values except $x \neq -3$ and the $x$ for which $f(x) \neq -3$
C. all real values except $x \neq 7$ and the $x$ for which $f(x) \neq 7$
D. all real values except $x \neq 7$ and the $x$ for which $f(x) \neq -3
Answer (2)
The domain of f ( x ) excludes x = 7 .
The domain of g ( x ) excludes x = − 3 .
The domain of ( g ∘ f ) ( x ) excludes x = 7 and values where f ( x ) = − 3 .
Therefore, the domain is all real values except x = 7 and the x for which f ( x ) = − 3 .
Explanation
Understanding the Problem We are given that the domain of f ( x ) is all real numbers except 7, and the domain of g ( x ) is all real numbers except -3. We want to find the domain of the composite function ( g ∘ f ) ( x ) = g ( f ( x )) . This means we need to consider two restrictions: first, x cannot be 7 because that's not in the domain of f ( x ) ; second, f ( x ) cannot be -3 because that's not in the domain of g ( x ) .
Considering the Domain of f(x) The domain of ( g ∘ f ) ( x ) consists of all x in the domain of f such that f ( x ) is in the domain of g . Since the domain of f ( x ) is all real numbers except x = 7 , we must exclude x = 7 from the domain of ( g ∘ f ) ( x ) .
Considering the Domain of g(x) Since the domain of g ( x ) is all real numbers except x = − 3 , we must exclude all x such that f ( x ) = − 3 from the domain of ( g ∘ f ) ( x ) . Therefore, the domain of ( g ∘ f ) ( x ) is all real numbers except x = 7 and the x values for which f ( x ) = − 3 .
Final Answer Thus, the domain of ( g ∘ f ) ( x ) is all real values except x = 7 and the x for which f ( x ) = − 3 .
Examples
Composition of functions is a fundamental concept in mathematics with numerous real-world applications. For instance, consider a scenario where a store offers a discount of 20% on all items, and there's an additional coupon for 10 o ff t h e d i sco u n t e d p r i ce . I f f(x) = 0.8x re p rese n t s t h e 20 g(x) = x - 10 re p rese n t s t h eco u p o n , t h e n (g \circ f)(x) = g(f(x)) = 0.8x - 10$ calculates the final price after both discounts. Understanding the domain of such composite functions ensures that the calculations are valid and meaningful, such as ensuring the price remains non-negative.
The domain of the composite function ( g ∘ f ) ( x ) is all real values except where x = 7 and the values for which f ( x ) = − 3 . Hence, the chosen option is A: all real values except x = − 3 and the x for which f ( x ) = 7 .
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