Consider the graphs of [tex]f(x)=x^3[/tex] and of [tex]g(x)=\frac{1}{x^3}[/tex]. Are the composite functions commutative? Why or why not?
A. They are not necessarily commutative because [tex]f(g(1))=g(f(1))[/tex].
B. They are not commutative; the composites do not simplify to [tex]x[/tex].
C. They are not commutative because the domains of [tex]f(x)[/tex] and [tex]g(x)[/tex] are different.
D. They are not commutative because the graphs intersect each other.
Answer (1)
Calculate the composite function f ( g ( x )) : f ( g ( x )) = f ( x 3 1 ) = ( x 3 1 ) 3 = x 9 1 .
Calculate the composite function g ( f ( x )) : g ( f ( x )) = g ( x 3 ) = ( x 3 ) 3 1 = x 9 1 .
Compare f ( g ( x )) and g ( f ( x )) : Since f ( g ( x )) = g ( f ( x )) = x 9 1 , the composite functions are equal.
Consider the domains: Both composite functions have the same domain, all real numbers except x = 0 . Therefore, the composite functions are commutative. The answer is that the composite functions are commutative.
Explanation
Understanding the Problem We are given two functions, f ( x ) = x 3 and g ( x ) = x 3 1 , and we want to determine if the composite functions f ( g ( x )) and g ( f ( x )) are commutative. In other words, we want to check if f ( g ( x )) = g ( f ( x )) for all x in their domains.
Calculating f(g(x)) First, let's find the composite function f ( g ( x )) . This means we substitute g ( x ) into f ( x ) : f ( g ( x )) = f ( x 3 1 ) = ( x 3 1 ) 3 = x 9 1 .
Calculating g(f(x)) Next, let's find the composite function g ( f ( x )) . This means we substitute f ( x ) into g ( x ) : g ( f ( x )) = g ( x 3 ) = ( x 3 ) 3 1 = x 9 1 .
Comparing the Composite Functions and Domains Now, we compare f ( g ( x )) and g ( f ( x )) . We found that f ( g ( x )) = x 9 1 and g ( f ( x )) = x 9 1 . Since f ( g ( x )) = g ( f ( x )) , the composite functions are equal. However, we must consider the domains of the functions. The domain of f ( x ) = x 3 is all real numbers. The domain of g ( x ) = x 3 1 is all real numbers except x = 0 , since we cannot divide by zero. For the composite function f ( g ( x )) , we have g ( x ) inside f ( x ) , so we must exclude x = 0 from the domain. For the composite function g ( f ( x )) , we have f ( x ) inside g ( x ) , so we must also exclude x = 0 from the domain, since f ( 0 ) = 0 and g ( 0 ) is undefined. Therefore, both composite functions have the same domain, which is all real numbers except x = 0 . Since f ( g ( x )) = g ( f ( x )) and they have the same domain, the composite functions are commutative.
Conclusion Therefore, the composite functions are commutative.
Examples
Understanding commutative properties of functions is crucial in fields like signal processing and cryptography. For instance, in image processing, applying a sequence of filters might yield different results depending on the order. If two filters, represented by functions f ( x ) and g ( x ) , are commutative, i.e., f ( g ( x )) = g ( f ( x )) , then the order in which they are applied doesn't matter, simplifying the processing pipeline. This concept extends to encryption algorithms, where the order of applying encryption and decryption functions can be critical for security.
