Suppose $f(x)=(5-x)^3$ and $g(x)=a+x^2$, and $f(g(x))=\left(1-x^2\right)^3$. What is the value of $a$?
A. $a=-6$
B. $a=-4$
C. $a=4$
D. $a=6
Answer (1)
Substitute g ( x ) into f ( x ) to find f ( g ( x )) = ( 5 − a − x 2 ) 3 .
Set the expression equal to the given f ( g ( x )) : ( 5 − a − x 2 ) 3 = ( 1 − x 2 ) 3 .
Take the cube root of both sides: 5 − a − x 2 = 1 − x 2 .
Solve for a : a = 4 .
Explanation
Understanding the Problem We are given the functions f ( x ) = ( 5 − x ) 3 , g ( x ) = a + x 2 , and f ( g ( x )) = ( 1 − x 2 ) 3 . Our goal is to find the value of a .
Finding f(g(x)) First, we need to find an expression for f ( g ( x )) . We substitute g ( x ) into f ( x ) :
f ( g ( x )) = f ( a + x 2 ) = ( 5 − ( a + x 2 ) ) 3 = ( 5 − a − x 2 ) 3 .
Equating the Expressions We are given that f ( g ( x )) = ( 1 − x 2 ) 3 . Therefore, we can set the two expressions for f ( g ( x )) equal to each other: ( 5 − a − x 2 ) 3 = ( 1 − x 2 ) 3 .
Taking the Cube Root Now, we take the cube root of both sides of the equation: 3 ( 5 − a − x 2 ) 3 = 3 ( 1 − x 2 ) 3 5 − a − x 2 = 1 − x 2 .
Simplifying the Equation Next, we simplify the equation by adding x 2 to both sides: 5 − a − x 2 + x 2 = 1 − x 2 + x 2 5 − a = 1.
Solving for a Finally, we solve for a by adding a to both sides and subtracting 1 from both sides: 5 − a + a − 1 = 1 + a − 1 4 = a .
Conclusion Therefore, the value of a is 4.
Examples
Imagine you are designing a security system where the final output needs to match a specific function. By understanding function composition, as demonstrated in this problem, you can adjust the parameters of the inner function (like 'a' in our problem) to ensure the overall system behaves as desired. This is crucial in engineering and programming where precise control over system behavior is essential. For example, if f ( x ) represents an encryption algorithm and g ( x ) represents a data transformation, you can tune g ( x ) to ensure the encrypted output f ( g ( x )) meets specific security requirements.
