Question

attempt 1: 10 attempts remaining. find a pair of functions ( f(x) ) and ( g(x) ) so the given function ( h(x) = \frac{6}{(x + 2)^3} ) can be expressed as ( h(x) = f(g(x)) ). (both answers must be correct to receive any credit.) ( f(x) = ) ( g(x) = ) submit answer next item

Explanation:

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

The function \( h(x)=\frac{6}{(x + 2)^3} \) is a composite function. The inner function \( g(x) \) is the part inside the more complex operation. Here, the expression inside the cube and the denominator is \( x+2 \), so we can take \( g(x)=x + 2 \).

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

Now, we need a function \( f \) such that when we substitute \( g(x) \) into \( f \), we get \( h(x) \). If \( g(x)=x + 2 \), then \( (g(x))^3=(x + 2)^3 \), and we have \( \frac{6}{(g(x))^3} \). So we can define \( f(x)=\frac{6}{x^3} \), because when we substitute \( x = g(x) \) into \( f \), we get \( f(g(x))=\frac{6}{(g(x))^3}=\frac{6}{(x + 2)^3}=h(x) \).

Answer:

\( f(x)=\frac{6}{x^3} \), \( g(x)=x + 2 \) (Note: There are other possible pairs, but this is a valid one.)