Question

Suppose f(x) = (5 – x)³ and g(x) = a + x², and f(g(x)) = (1 – x²)³. What is the value of a?

A. a = –6
B. a = –4
C. a = 4
D. a = 6

Answer 1

By composing the functions f(x) and g(x), and setting the result equal to (1 - x²)³, we simplify the expression and cancel out x² terms. Solving the equation for a, we find that the value of a is 4.

Step-by-step explanation:

To determine the value of a in the function g(x) = a + x², we need to compose the functions f(x) and g(x) and compare the result with (1 – x²)³, as given by the equation f(g(x)) = (1 – x²)³. We start by composing f(x) = (5 – x)³ with g(x) to obtain f(g(x)) = (5 - (a + x²))³.

Setting the composed function equal to (1 – x²)³ gives us:

(5 – (a + x²))³ = (1 – x²)³.

To find a, we recognize that the two expressions must be equivalent for all values of x. This means that 5 – (a + x²) must equal 1 – x². Simplifying further:

5 – a – x² = 1 – x²

Since the x² terms cancel out, what we are left with is:

5 – a = 1,

Solving this equation for a:

a = 5 – 1 = 4.

Therefore, the value of a is 4.