Question

find two functions f and g such that (f ∘ g)(x) = h(x). (there are many correct answers. use non-identity functions for f(x) and g(x).) h(x) = (6 - x)³ (f(x), g(x)) = ( )

Explanation:

Step1: Identify the inner function \( g(x) \)

The function \( h(x) = (6 - x)^3 \) is a composition of two functions. The inner function \( g(x) \) can be the expression inside the cube, so let \( g(x)=6 - x \).

Step2: Identify the outer function \( f(x) \)

The outer function \( f(x) \) should take the output of \( g(x) \) and cube it. So let \( f(x)=x^3 \).

Step3: Verify the composition

Now, we check the composition \( (f\circ g)(x) \). By the definition of composition, \( (f\circ g)(x)=f(g(x)) \). Substitute \( g(x) = 6 - x \) into \( f(x) \), we get \( f(6 - x)=(6 - x)^3 \), which is equal to \( h(x) \).

Answer:

\( f(x)=x^3 \), \( g(x)=6 - x \) (Note: There are other possible answers, for example, \( f(x)=(x)^3 \) and \( g(x) = 6 - x \) is a valid pair of functions for the composition.)