ION 4 - 1 POINT the function \( h(x) \) below, select the answer choice which correctly decomposes \( h(x) \) into component functions \( f \) ( \( (x) \) so that \( h(x) \Rightarrow f(g(x)) \). \[ h(x)=(-x+6)^{2} \] the correct answer below: \( (x)=f(g(x)) \), where \( f(x)=-x \) and \( g(x)=(x+6)^{2} \) \( (x)=f(g(x)) \), where \( f(x)=-x+6 \) and \( g(x)=x^{2} \)

To decompose the function \( h(x) = (-x + 6)^2 \) into component functions \( f \) and \( g \) such that \( h(x) = f(g(x)) \), we need to identify suitable functions \( f \) and \( g \). 1. **Identify the structure of \( h(x) \)**: - The function \( h(x) \) is a composition of two operations: first, we apply the linear transformation \( -x + 6 \), and then we square the result. 2. **Choose \( g(x) \)**: - A natural choice for \( g(x) \) is the inner function that we apply first. In this case, we can let: \[ g(x) = -x + 6 \] 3. **Choose \( f(x) \)**: - The outer function \( f(x) \) will then be the operation we apply to \( g(x) \). Since we are squaring the result of \( g(x) \), we can define: \[ f(x) = x^2 \] 4. **Combine \( f \) and \( g \)**: - Now we can express \( h(x) \) as: \[ h(x) = f(g(x)) = f(-x + 6) = (-x + 6)^2 \] Now, let's check the provided answer choices: 1. **First choice**: \( f(x) = -x \) and \( g(x) = (x + 6)^2 \) - This does not match our decomposition since \( g(x) \) should not be squared. 2. **Second choice**: \( f(x) = -x + 6 \) and \( g(x) = x^2 \) - This also does not match our decomposition since \( g(x) \) should not be \( x^2 \). Neither of the provided answer choices correctly decomposes \( h(x) \) into \( f(g(x)) \). The correct decomposition is: - \( g(x) = -x + 6 \) - \( f(x) = x^2 \) Thus, the correct answer is not listed among the choices provided.