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)
Determine the composite functions f ( g ( x )) = ∣ x 3 ∣ 1 + 1 and g ( f ( x )) = ( ∣ x ∣ + 1 ) 3 1 .
Analyze the given options and consider the case when x < 0 .
Observe that when x < 0 , g ( x ) takes on negative values, which affects the absolute value in f ( x ) .
Conclude that the composite functions are not commutative because the graph of g ( x ) contains negative values. T h e g r a p h o f g ( x ) co n t ain s n e g a t i v e v a l u es .
Explanation
Analyzing the Functions Let's analyze the given functions and the options to determine why the composite functions f ( g ( x )) and g ( f ( x )) are not commutative.
The functions are f ( x ) = ∣ x ∣ + 1 and g ( x ) = x 3 1 . We want to find the reason why f ( g ( x )) = g ( f ( x )) .
Finding Composite Functions First, let's find the composite functions:
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
Analyzing the Options Now, let's analyze the given options:
The graphs of f ( x ) and g ( x ) intersect each other. This is not directly related to the commutativity of composite functions.
The graph of g ( x ) contains negative values. This is relevant because when x < 0 , g ( x ) < 0 , and the absolute value in f ( g ( x )) will make it positive. This difference in sign behavior can lead to non-commutativity.
The domains of f ( x ) and g ( x ) are different. The domain of f ( x ) is all real numbers, and the domain of g ( x ) is all real numbers except x = 0 . This difference in domains can also contribute to non-commutativity.
The y -intercepts of f ( x ) and g ( x ) are different. The y -intercept of f ( x ) is f ( 0 ) = ∣0∣ + 1 = 1 . The function g ( x ) does not have a y -intercept because it is not defined at x = 0 . This difference in y -intercepts is not the primary reason for non-commutativity.
Considering a Negative Value of x Let's consider the case when x < 0 . For example, let x = − 1 .
f ( g ( − 1 )) = ∣ ( − 1 ) 3 ∣ 1 + 1 = ∣ − 1∣ 1 + 1 = 1 + 1 = 2
g ( f ( − 1 )) = ( ∣ − 1∣ + 1 ) 3 1 = ( 1 + 1 ) 3 1 = 2 3 1 = 8 1
Since f ( g ( − 1 )) = 2 and g ( f ( − 1 )) = 8 1 , we can see that f ( g ( x )) = g ( f ( x )) when x < 0 . This is because g ( x ) takes on negative values when x < 0 , and the absolute value in f ( x ) affects the result.
The Key Observation The key observation is that the graph of g ( x ) contains negative values. This is because when x < 0 , g ( x ) = x 3 1 < 0 . The absolute value in f ( x ) = ∣ x ∣ + 1 affects the composite function f ( g ( x )) differently than it affects g ( f ( x )) .
Final Answer Therefore, the composite functions f ( g ( x )) and g ( f ( x )) are not commutative because the graph of g ( x ) contains negative values.
Examples
Understanding function composition is crucial in many areas of mathematics and computer science. For instance, in signal processing, you might have a function that filters noise from a signal, and another function that amplifies the signal. The order in which you apply these functions matters; filtering before amplifying might yield a different result than amplifying before filtering. Similarly, in cryptography, encryption and decryption functions must be applied in the correct order to ensure secure communication. The non-commutativity of composite functions highlights the importance of carefully considering the order of operations in these and other applications.
