Consider the graphs of $f(x)=|x|+1$ and $g(x)=\frac{1}{x^3}$. The composite functions $f(g(x))$ and $g(f(x))$ are not commutative, based on which observation?
A. The graphs of $f(x)$ and $g(x)$ intersect each other.
B. The graph of $g(x)$ contains negative values.
C. The domains of $f(x)$ and $g(x)$ are different.
D. The $y$-intercepts of $f(x)$ and $g(x)$ are different.
Answer (1)
Calculate the composite functions: f ( g ( x )) = ∣ x 3 ∣ 1 + 1 and g ( f ( x )) = ( ∣ x ∣ + 1 ) 3 1 .
Analyze the given options and determine their relevance to the non-commutativity of the composite functions.
Identify that the graph of g ( x ) contains negative values, which contributes to the non-commutativity because f ( x ) is always positive.
Conclude that the composite functions are not commutative because 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
Understanding the Problem We are given two functions, f ( x ) = ∣ x ∣ + 1 and g ( x ) = x 3 1 , and 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 )) . We are given four possible observations and must choose the correct one.
Finding Composite Functions First, 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
We can see that f ( g ( x )) = ∣ x 3 ∣ 1 + 1 and g ( f ( x )) = ( ∣ 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 non-commutativity of composite functions.
The graph of g ( x ) contains negative values. This is a key observation. When x < 0 , g ( x ) = x 3 1 < 0 . Since f ( x ) = ∣ x ∣ + 1 , f ( x ) is always positive. This difference in sign behavior contributes to the 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 . While this is true, it doesn't directly explain why f ( g ( x )) = g ( f ( x )) .
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 doesn't directly explain the non-commutativity.
Determining the Correct Observation The most relevant observation is that the graph of g ( x ) contains negative values. This is because when x < 0 , g ( x ) is negative, while f ( x ) is always positive. This difference in sign behavior contributes to the non-commutativity of the composite functions.
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
Consider a scenario where you have a temperature sensor f ( x ) = ∣ x ∣ + 1 that always returns a positive temperature (absolute value plus 1) and a cooling system g ( x ) = x 3 1 that can either cool or heat depending on the input. If you first measure the temperature and then apply the cooling system, g ( f ( x )) , you get a different result than if you first apply the cooling system and then measure the temperature, f ( g ( x )) . This is because the cooling system can produce negative values, which the temperature sensor then converts to positive values, leading to different outcomes. This non-commutativity highlights how the order of operations matters when dealing with systems that can produce both positive and negative values.
