For which pair of functions is $(g \circ f)(a)=|a|-2$?
A. $f(a)=a^2-4$ and $g(a)=\sqrt{a}$
B. $f(a)=\frac{1}{2} a-1$ and $g(a)=2 a-2$
C. $f(a)=5+a^2$ and $g(a)=\sqrt{a-5}-2$
D. $f(a)=3-3 a$ and $g(a)=4 a-5$
Answer (2)
The pair of functions that satisfies the equation ( g ∘ f ) ( a ) = ∣ a ∣ − 2 is f ( a ) = 5 + a 2 and g ( a ) = a − 5 − 2 . This was determined by evaluating each pair and identifying the one that matched the required form. Hence, the answer is option C.
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Evaluate ( g ∘ f ) ( a ) for each pair of functions.
Pair 1: ( g ∘ f ) ( a ) = a 2 − 4 = ∣ a ∣ − 2 .
Pair 2: ( g ∘ f ) ( a ) = a − 4 = ∣ a ∣ − 2 .
Pair 3: ( g ∘ f ) ( a ) = ∣ a ∣ − 2 .
Pair 4: ( g ∘ f ) ( a ) = 7 − 12 a = ∣ a ∣ − 2 .
The correct pair is f ( a ) = 5 + a 2 and g ( a ) = a − 5 − 2 , so the answer is f ( a ) = 5 + a 2 and g ( a ) = a − 5 − 2 .
Explanation
Understanding the Problem We are given four pairs of functions f ( a ) and g ( a ) , and we need to find the pair for which ( g ∘ f ) ( a ) = ∣ a ∣ − 2 . The composition ( g ∘ f ) ( a ) means g ( f ( a )) . We will evaluate each pair to see which one satisfies the given condition.
Evaluating Pair 1 For the first pair, f ( a ) = a 2 − 4 and g ( a ) = a . Then ( g ∘ f ) ( a ) = g ( f ( a )) = g ( a 2 − 4 ) = a 2 − 4 . This is not equal to ∣ a ∣ − 2 for all a . For example, if a = 0 , then a 2 − 4 = − 4 , which is not a real number, while ∣0∣ − 2 = − 2 .
Evaluating Pair 2 For the second pair, f ( a ) = 2 1 a − 1 and g ( a ) = 2 a − 2 . Then ( g ∘ f ) ( a ) = g ( f ( a )) = g ( 2 1 a − 1 ) = 2 ( 2 1 a − 1 ) − 2 = a − 2 − 2 = a − 4 . This is not equal to ∣ a ∣ − 2 for all a . For example, if a = 1 , then a − 4 = 1 − 4 = − 3 , while ∣1∣ − 2 = 1 − 2 = − 1 .
Evaluating Pair 3 For the third pair, f ( a ) = 5 + a 2 and g ( a ) = a − 5 − 2 . Then ( g ∘ f ) ( a ) = g ( f ( a )) = g ( 5 + a 2 ) = ( 5 + a 2 ) − 5 − 2 = a 2 − 2 = ∣ a ∣ − 2 . This is equal to ∣ a ∣ − 2 for all a .
Evaluating Pair 4 For the fourth pair, f ( a ) = 3 − 3 a and g ( a ) = 4 a − 5 . Then ( g ∘ f ) ( a ) = g ( f ( a )) = g ( 3 − 3 a ) = 4 ( 3 − 3 a ) − 5 = 12 − 12 a − 5 = 7 − 12 a . This is not equal to ∣ a ∣ − 2 for all a . For example, if a = 0 , then 7 − 12 a = 7 , while ∣0∣ − 2 = − 2 .
Final Answer Therefore, the pair of functions for which ( g ∘ f ) ( a ) = ∣ a ∣ − 2 is f ( a ) = 5 + a 2 and g ( a ) = a − 5 − 2 .
Examples
Composition of functions is a fundamental concept in mathematics and has many real-world applications. For example, consider a store that marks down all of its items by 10% and then applies a coupon for 5 o ff . I f f(x) = 0.9x re p rese n t s t h e ma r k d o w nan d g(x) = x - 5 re p rese n t s t h eco u p o n , t h e n (g \circ f)(x) = 0.9x - 5$ represents the final price of an item. Understanding function composition allows us to model and analyze such situations effectively.
