Consider the graphs of [tex]f(x)=|x|+1[/tex] and [tex]g(x)=\frac{1}{x^3}[/tex]. The composite functions [tex]f(g(x))[/tex] and [tex]g(f(x))[/tex] are not commutative, based on which observation?
A. The graphs of [tex]f(x)[/tex] and [tex]g(x)[/tex] intersect each other.
B. The graph of [tex]g(x)[/tex] contains negative values.
C. The domains of [tex]f(x)[/tex] and [tex]g(x)[/tex] are different.
D. The [tex]y[/tex]-intercepts of [tex]f(x)[/tex] and [tex]g(x)[/tex] are different.
Answer (1)
Find the composite functions f ( g ( x )) and g ( f ( x )) .
Determine the domains of f ( x ) , g ( x ) , f ( g ( x )) , and g ( f ( x )) .
Observe that the domain of f ( g ( x )) is ( − ∞ , 0 ) ∪ ( 0 , ∞ ) , while the domain of g ( f ( x )) is ( − ∞ , ∞ ) .
Conclude that since the domains of f ( g ( x )) and g ( f ( x )) are different, the composite functions are not commutative. The final answer is: The domains of f ( x ) and g ( x ) are different.
Explanation
Understanding the Problem We are given two functions, f ( x ) = ∣ x ∣ + 1 and g ( x ) = x 3 1 . We want to determine why the composite functions f ( g ( x )) and g ( f ( x )) are not commutative. This means we want to find out why f ( g ( x )) = g ( f ( x )) .
Finding Composite Functions Let's find the composite functions f ( g ( x )) and g ( f ( x )) .
f ( g ( x )) = f ( x 3 1 ) = x 3 1 + 1 = ∣ x 3 ∣ 1 + 1
g ( f ( x )) = g ( ∣ x ∣ + 1 ) = ( ∣ x ∣ + 1 ) 3 1
So, f ( g ( x )) = ∣ x 3 ∣ 1 + 1 and g ( f ( x )) = ( ∣ x ∣ + 1 ) 3 1 .
Analyzing Domains Now, let's analyze the domains of the functions.
The domain of f ( x ) = ∣ x ∣ + 1 is all real numbers, ( − ∞ , ∞ ) .
The domain of g ( x ) = x 3 1 is all real numbers except x = 0 , which is ( − ∞ , 0 ) ∪ ( 0 , ∞ ) .
The domain of f ( g ( x )) = ∣ x 3 ∣ 1 + 1 is all real numbers except x = 0 , which is ( − ∞ , 0 ) ∪ ( 0 , ∞ ) .
The domain of g ( f ( x )) = ( ∣ x ∣ + 1 ) 3 1 is all real numbers, ( − ∞ , ∞ ) , since ∣ x ∣ + 1 is always positive and never zero.
Conclusion Since the domain of f ( g ( x )) is ( − ∞ , 0 ) ∪ ( 0 , ∞ ) and the domain of g ( f ( x )) is ( − ∞ , ∞ ) , the domains of the two composite functions are different. This is enough to conclude that the two composite functions are not commutative.
Final Answer The reason why f ( g ( x )) and g ( f ( x )) are not commutative is that their domains are different. Specifically, f ( g ( x )) is not defined at x = 0 , while g ( f ( x )) is defined for all real numbers.
Examples
Understanding function composition and domains is crucial in many areas of mathematics and computer science. For example, in signal processing, composing functions can represent how a signal is transformed through different stages of a system. If the order of these transformations matters (i.e., the functions don't commute), the final output will be different. Similarly, in cryptography, the order in which encryption and decryption functions are applied is critical for security.
