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 (2)
Calculate f ( g ( x )) : Since f ( x ) = 1 , f ( g ( x )) = f ( 2 ) = 1 .
Calculate g ( f ( x )) : Since g ( x ) = 2 , g ( f ( x )) = g ( 1 ) = 2 .
Compare the results: f ( g ( x )) = 1 and g ( f ( x )) = 2 .
Conclude: Since 1 e q 2 , the compositions are not commutative: They are not commutative, because f ( g ( x )) and g ( f ( x )) are not equal.
Explanation
Understanding the Problem We are given two functions, f ( x ) = 1 and g ( x ) = 2 . We need to determine if their compositions are commutative, meaning whether f ( g ( x )) = g ( f ( x )) for all x .
Calculating f(g(x)) First, let's find f ( g ( x )) . Since f ( x ) = 1 for any input x , we have f ( g ( x )) = f ( 2 ) = 1 .
Calculating g(f(x)) Next, let's find g ( f ( x )) . Since g ( x ) = 2 for any input x , we have g ( f ( x )) = g ( 1 ) = 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 , the compositions are not commutative.
Final Answer Therefore, the compositions of f ( x ) = 1 and g ( x ) = 2 are not commutative because f ( g ( x )) e q g ( f ( x )) . The correct answer is: They are not commutative, because f ( g ( x )) and g ( f ( x )) are not equal.
Examples
In real life, commutativity can be seen in everyday actions. For example, putting on socks and then shoes is not commutative with putting on shoes and then socks; the order matters. Similarly, in mathematics, some operations depend on the order in which they are performed. Understanding commutativity helps in various fields, such as computer science (order of operations in algorithms) and physics (order of transformations).
The compositions of the functions f ( x ) = 1 and g ( x ) = 2 are not commutative because f ( g ( x )) = 1 and g ( f ( x )) = 2 are not equal. The correct answer is option D: they are not commutative, because the compositions are not equal.
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