Questions: Functions Expressing a function as a composition of two functions Suppose (H(x)=sqrt[3]2-5 x). Find two functions (f) and (g) such that ((f circ 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 given the function \( H(x) = \sqrt[3]{2 - 5x} \). We need to express \( H(x) \) as a composition of two functions \( f \) and \( g \), such that \( (f \circ g)(x) = H(x) \). Neither \( f \) nor \( g \) can be the identity function.

The function \( H(x) = \sqrt[3]{2 - 5x} \) can be thought of as applying two operations:

1. A linear transformation: \( g(x) = 2 - 5x \).
2. A cube root function: \( f(x) = \sqrt[3]{x} \).

Let:

• \( g(x) = 2 - 5x \).
• \( f(x) = \sqrt[3]{x} \).

Now, verify that \( (f \circ g)(x) = H(x) \): \[ (f \circ g)(x) = f(g(x)) = f(2 - 5x) = \sqrt[3]{2 - 5x} = H(x). \]

Final Answer