Questions: Suppose H(x)=3 √x+3. 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.) f(x)= g(x)=

Solution Steps

We are tasked with finding two functions \( f \) and \( g \) such that \( (f \circ g)(x) = H(x) \), where \( H(x) = 3\sqrt{x} + 3 \). The composition \( (f \circ g)(x) \) means \( f(g(x)) \). Neither \( f \) nor \( g \) can be the identity function.

The function \( H(x) = 3\sqrt{x} + 3 \) can be broken down into two parts:

1. The inner function \( g(x) = \sqrt{x} \).
2. The outer function \( f(x) = 3x + 3 \).

Let's verify that \( f(g(x)) = H(x) \): \[ f(g(x)) = f(\sqrt{x}) = 3\sqrt{x} + 3 = H(x). \] This satisfies the condition \( (f \circ g)(x) = H(x) \).

Final Answer