Are the compositions of [tex]f(x)=1[/tex] and [tex]g(x)=2[/tex] commutative? Why or why not?
A. They are commutative, because [tex]f(x)[/tex] and [tex]g(x)[/tex] are constant functions.
B. They are commutative, because [tex]f(g(x))[/tex] and [tex]g(f(x))[/tex] are constant functions.
C. They are not commutative, because [tex]f(x)[/tex] and [tex]g(x)[/tex] are not equal.
D. They are not commutative, because [tex]f(g(x))[/tex] and [tex]g(f(x))[/tex] are not equal.
Answer (1)
Calculate f ( g ( x )) : Since g ( x ) = 2 , f ( g ( x )) = f ( 2 ) = 1 .
Calculate g ( f ( x )) : Since f ( x ) = 1 , g ( f ( x )) = g ( 1 ) = 2 .
Compare the results: f ( g ( x )) = 1 and g ( f ( x )) = 2 .
Conclude that the compositions are not commutative because 1 e q 2 , so the answer is They are not commutative, because f ( g ( x )) and g ( f ( x )) are not equal.
Explanation
Understanding the Problem We are asked to determine whether the composition of two constant functions, f ( x ) = 1 and g ( x ) = 2 , is commutative. In other words, we need to check if f ( g ( x )) = g ( f ( x )) for all x .
Calculating f(g(x)) First, let's find f ( g ( x )) . Since g ( x ) = 2 for any x , we have f ( g ( x )) = f ( 2 ) . Because f ( x ) = 1 for any x , f ( 2 ) = 1 . Thus, f ( g ( x )) = 1 .
Calculating g(f(x)) Next, let's find g ( f ( x )) . Since f ( x ) = 1 for any x , we have g ( f ( x )) = g ( 1 ) . Because g ( x ) = 2 for any x , g ( 1 ) = 2 . Thus, g ( f ( x )) = 2 .
Comparing the Results Now, we compare f ( g ( x )) and g ( f ( x )) . We found that f ( g ( x )) = 1 and g ( f ( x )) = 2 . Since 1 e q 2 , f ( g ( x )) e q g ( f ( x )) . Therefore, the composition of f ( x ) and g ( x ) is not commutative.
Final Answer The compositions of f ( x ) = 1 and g ( x ) = 2 are not commutative because f ( g ( x )) = 1 and g ( f ( x )) = 2 , so f ( g ( x )) e q g ( f ( x )) .
Examples
In real life, function composition can model sequential processes. For example, consider a discount coupon applied to a product. If f(x) represents a 10% discount and g(x) represents a $5 off coupon, then f(g(x)) means applying the $5 off coupon first, then taking 10% off the reduced price. Conversely, g(f(x)) means taking 10% off first, then applying the $5 off coupon. The final price will differ depending on the order, demonstrating non-commutative behavior. Understanding function composition helps in optimizing such sequential operations.
