Which statement describes function composition with respect to the commutative property?
A. Given [tex]f(x)=x^2-4[/tex] and [tex]g(x)=x-3[/tex], [tex](f \cdot g)(2)=-3[/tex] and [tex](g \cdot f(2)=-3[/tex], so function composition is commutative.
B. Given [tex]f(x)=2 x-5[/tex] and [tex]g(x)=0.5 x-2.5[/tex], [tex](f \cdot g)(x)=x[/tex] and [tex](g \cdot f)(x)=x[/tex], so function composition is commutative.
C. Given [tex]f(x)=x^2[/tex] and [tex]g(x)=\sqrt{x}[/tex], [tex](f \cdot g)(4)=16[/tex] and [tex](g \cdot f)(4)=2[/tex], so function composition is not commutative.
D. Given [tex]f(x)=4 x[/tex] and [tex]g(x)=x^2[/tex], [tex](f \cdot g)(x)=4 x^2[/tex] and [tex](g \cdot f(x)=16 x^2[/tex], so function composition is not commutative.
Answer (2)
The correct statement regarding function composition and the commutative property is Statement D. It correctly shows that for f ( x ) = 4 x and g ( x ) = x 2 , the two compositions yield different results, indicating that function composition is not commutative. Therefore, the answer is D.
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Function composition is commutative if ( f c i rc g ) ( x ) = ( g c i rc f ) ( x ) for all x.
Statement 1 is incorrect because equality at one point doesn't imply general commutativity.
Statement 2 is incorrect because ( f c i rc g ) ( x ) e q ( g c i rc f ) ( x ) .
Statement 3 has incorrect values but correctly concludes non-commutativity.
Statement 4 correctly states that since ( f c i rc g ) ( x ) e q ( g c i rc f ) ( x ) , function composition is not commutative. Therefore, the answer is: G i v e n f ( x ) = 4 x an d g ( x ) = x 2 , ( f ⋅ g ) ( x ) = 4 x 2 an d ( g ⋅ f ) ( x ) = 16 x 2 , so f u n c t i o n co m p os i t i o ni s n o t co mm u t a t i v e .
Explanation
Understanding Commutativity We need to determine which statement accurately describes function composition with respect to the commutative property. Function composition is commutative if ( f c i rc g ) ( x ) = ( g c i rc f ) ( x ) for all x in the domain. We will analyze each statement to see if its conclusion about commutativity is correct.
Analyzing Statement 1 Statement 1: Given f ( x ) = x 2 − 4 and g ( x ) = x − 3 , the statement says ( f c i rc g ) ( 2 ) = − 3 and ( g c i rc f ) ( 2 ) = − 3 , so function composition is commutative. However, just because the composition is equal at one point does not mean it is commutative in general. From the tool, we have ( f c i rc g ) ( x ) = ( x − 3 ) 2 − 4 = x 2 − 6 x + 5 and ( g c i rc f ) ( x ) = x 2 − 4 − 3 = x 2 − 7 . Since these are not equal, the statement is incorrect.
Analyzing Statement 2 Statement 2: Given f ( x ) = 2 x − 5 and g ( x ) = 0.5 x − 2.5 , the statement says ( f c i rc g ) ( x ) = x and ( g c i rc f ) ( x ) = x , so function composition is commutative. From the tool, we have ( f c i rc g ) ( x ) = 2 ( 0.5 x − 2.5 ) − 5 = x − 5 − 5 = x − 10 and ( g c i rc f ) ( x ) = 0.5 ( 2 x − 5 ) − 2.5 = x − 2.5 − 2.5 = x − 5 . Since ( f c i rc g ) ( x ) e q ( g c i rc f ) ( x ) , the statement is incorrect.
Analyzing Statement 3 Statement 3: Given f ( x ) = x 2 and g ( x ) = x , the statement says ( f c i rc g ) ( 4 ) = 16 and ( g c i rc f ) ( 4 ) = 2 , so function composition is not commutative. From the tool, we have ( f c i rc g ) ( x ) = ( x ) 2 = x and ( g c i rc f ) ( x ) = x 2 = ∣ x ∣ . Then ( f c i rc g ) ( 4 ) = 4 and ( g c i rc f ) ( 4 ) = 4 2 = 4 . The values given in the statement are incorrect, but the conclusion that the composition is not commutative is correct since x and ∣ x ∣ are not the same for all x (e.g., when x < 0 ).
Analyzing Statement 4 Statement 4: Given f ( x ) = 4 x and g ( x ) = x 2 , the statement says ( f c i rc g ) ( x ) = 4 x 2 and ( g c i rc f ) ( x ) = 16 x 2 , so function composition is not commutative. From the tool, we have ( f c i rc g ) ( x ) = 4 ( x 2 ) = 4 x 2 and ( g c i rc f ) ( x ) = ( 4 x ) 2 = 16 x 2 . Since ( f c i rc g ) ( x ) e q ( g c i rc f ) ( x ) , the statement is correct.
Conclusion Therefore, the statement that correctly describes function composition with respect to the commutative property is: Given f ( x ) = 4 x and g ( x ) = x 2 , ( f c i rc g ) ( x ) = 4 x 2 and ( g c i rc f ) ( x ) = 16 x 2 , so function composition is not commutative.
Examples
Function composition is a fundamental concept in mathematics and computer science. In real life, it can be seen in manufacturing processes. For example, consider a car assembly line. One function might be installing the engine, e ( x ) , and another function might be painting the car, p ( x ) . The order in which these operations are performed matters. If you first install the engine and then paint the car, p ( e ( x )) , it's different from painting the car first and then trying to install the engine, e ( p ( x )) . This illustrates that, like function composition, the order of operations can significantly impact the final outcome.
