Questions: Suppose H(x)=(4x+8)^2. Find two functions f and g such that (f ∘ g)(x)=H(x). Neither function can be the identity function. (There may be more than one correct answer.)

Solution Steps

To solve this problem, we need to express the function \( H(x) = (4x + 8)^2 \) as a composition of two functions \( f \) and \( g \) such that \( (f \circ g)(x) = H(x) \). We can choose \( g(x) \) to be the inner function and \( f(x) \) to be the outer function. A natural choice is to let \( g(x) = 4x + 8 \) and \( f(x) = x^2 \), so that \( f(g(x)) = (4x + 8)^2 \).

We need to express the function \( H(x) = (4x + 8)^2 \) as a composition of two functions \( f \) and \( g \). We can choose: \[ g(x) = 4x + 8 \] \[ f(x) = x^2 \]

To verify that our choice of functions is correct, we compute the composition \( (f \circ g)(x) \): \[ (f \circ g)(x) = f(g(x)) = f(4x + 8) = (4x + 8)^2 \] This confirms that our functions satisfy the requirement \( (f \circ g)(x) = H(x) \).

Now, we will calculate \( H(5) \): \[ g(5) = 4(5) + 8 = 20 + 8 = 28 \] \[ H(5) = f(g(5)) = f(28) = 28^2 = 784 \]

Final Answer