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) = (4x + 6)² (f(x), g(x)) = (\boxed{ }, )

Explanation:

Step1: Identify the inner function

The composite function \( h(x)=(4x + 6)^2 \) can be seen as a function composition. The inner function \( g(x) \) should be the part inside the square, so let \( g(x)=4x + 6 \).

Step2: Identify the outer function

The outer function \( f(x) \) should be the function that takes the output of \( g(x) \) and squares it. So let \( f(x)=x^2 \).

Step3: Verify the composition

Now, we check the composition \( (f\circ g)(x)=f(g(x)) \). Substitute \( g(x) \) into \( f(x) \): \( f(4x + 6)=(4x + 6)^2 \), which is equal to \( h(x) \).

Answer:

\( f(x)=x^2 \), \( g(x)=4x + 6 \) (So the pair is \( (x^2, 4x + 6) \))