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 (2)
Calculate the composite function f ( g ( x )) : f ( g ( x )) = f ( x 3 1 ) = x 9 1 .
Calculate the composite function g ( f ( x )) : g ( f ( x )) = g ( x 3 ) = x 9 1 .
Compare the two composite functions: f ( g ( x )) = g ( f ( x )) .
Conclude that the composite functions are commutative: commutative .
Explanation
Understanding the Problem We are given two functions, f ( x ) = x 3 and g ( x ) = x 3 1 , and we need to determine if their composite functions are commutative. Two functions are commutative 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 )) . 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 )) . We substitute f ( x ) into g ( x ) : g ( f ( x )) = g ( x 3 ) = ( x 3 ) 3 1 = x 9 1 .
Comparing the Composite Functions 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 )) for all x in their domains (except x = 0 , where g ( x ) is undefined), the composite functions are commutative.
Conclusion Therefore, the composite functions are commutative because f ( g ( x )) = g ( f ( x )) = x 9 1 .
Examples
In physics, understanding commutative operations is crucial when dealing with transformations. For example, if two rotations are commutative, the order in which they are applied does not affect the final orientation. This concept is also applicable in signal processing, where the order of applying filters might or might not change the final output depending on whether the filters commute.
The composite functions f ( g ( x )) and g ( f ( x )) are commutative because they produce the same result of x 9 1 . Thus, they satisfy the commutative property for their respective domains (excluding zero).
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