Which pair of functions represents a decomposition of $f(g(x))=\left|2(x+1)^2+(x+1)\right|$?
A. $f(x)=(x+1)^2$ and $g(x)=|2 x+1|$
B. $f(x)=(x+1)$ and $g(x)=\left|2 x^2+x\right|$
C. $f(x)=|2 x+1|$ and $g(x)=(x+1)^2$
D. $f(x)=\left|2 x^2+x\right|$ and $g(x)=(x+1)$
Answer (1)
We are given f ( g ( x )) = ∣2 ( x + 1 ) 2 + ( x + 1 ) ∣ and need to find f ( x ) and g ( x ) .
Test each option by computing f ( g ( x )) .
Option 1: f ( x ) = ( x + 1 ) 2 and g ( x ) = ∣2 x + 1∣ gives f ( g ( x )) = ( ∣2 x + 1∣ + 1 ) 2 , which doesn't match.
Option 2: f ( x ) = ( x + 1 ) and g ( x ) = ∣2 x 2 + x ∣ gives f ( g ( x )) = ∣2 x 2 + x ∣ + 1 , which doesn't match.
Option 3: f ( x ) = ∣2 x + 1∣ and g ( x ) = ( x + 1 ) 2 gives f ( g ( x )) = ∣2 ( x + 1 ) 2 + 1∣ , which doesn't match.
Option 4: f ( x ) = ∣2 x 2 + x ∣ and g ( x ) = ( x + 1 ) gives f ( g ( x )) = ∣2 ( x + 1 ) 2 + ( x + 1 ) ∣ , which matches.
The correct pair is f ( x ) = ∣2 x 2 + x ∣ and g ( x ) = ( x + 1 ) .
Explanation
Understanding the Problem We are given a composite function f ( g ( x )) = ∣2 ( x + 1 ) 2 + ( x + 1 ) ∣ . We need to find the pair of functions f ( x ) and g ( x ) such that when we plug g ( x ) into f ( x ) , we get the given composite function. We will test each option by computing f ( g ( x )) and comparing it to the given expression.
Testing Option 1 Let's test the first option: f ( x ) = ( x + 1 ) 2 and g ( x ) = ∣2 x + 1∣ . Then f ( g ( x )) = f ( ∣2 x + 1∣ ) = ( ∣2 x + 1∣ + 1 ) 2 . This does not match the given composite function.
Testing Option 2 Now let's test the second option: f ( x ) = ( x + 1 ) and g ( x ) = ∣2 x 2 + x ∣ . Then f ( g ( x )) = f ( ∣2 x 2 + x ∣ ) = ∣2 x 2 + x ∣ + 1 . This also does not match the given composite function.
Testing Option 3 Let's test the third option: f ( x ) = ∣2 x + 1∣ and g ( x ) = ( x + 1 ) 2 . Then f ( g ( x )) = f (( x + 1 ) 2 ) = ∣2 ( x + 1 ) 2 + 1∣ . This does not match the given composite function.
Testing Option 4 Finally, let's test the fourth option: f ( x ) = ∣2 x 2 + x ∣ and g ( x ) = ( x + 1 ) . Then f ( g ( x )) = f ( x + 1 ) = ∣2 ( x + 1 ) 2 + ( x + 1 ) ∣ . This matches the given composite function.
Conclusion Therefore, the pair of functions that represents the decomposition of f ( g ( x )) = ∣2 ( x + 1 ) 2 + ( x + 1 ) ∣ is f ( x ) = ∣2 x 2 + x ∣ and g ( x ) = ( x + 1 ) .
Examples
Composite functions are used in many real-world applications, such as in computer graphics to apply multiple transformations to an object (e.g., scaling, rotation, and translation). Each transformation can be represented as a function, and the composite function represents the combined effect of applying these transformations in sequence. Understanding function decomposition helps in designing modular and reusable code.
