Question

1. move each item into the correct order. once all items are in the correct order, submit your response.

list the steps in order to decompose the function ( h(x) = (x - 5)^2 ), where ( h(x) = g(f(x)) ).

• find what ( g(x) ) would have to be to show a squared function. ( g(x) = x^2 )
• looking at ( h(x) ), notice the function is squared and see that the exponent 2 is being applied to ( (x - 5) )
• since ( g(x) = x^2 ), ( f(x) ) can now be found. ( f(x) = x - 5 )
• check the composition of ( f(x) = x - 5 ) and ( g(x) = x^2 ) to verify ( h(x) = (x - 5)^2 ). ( g(f(x)) = g(x - 5) = (x - 5)^2 = h(x) )

Explanation:

Step 1: Analyze the function structure

First, we examine \( h(x)=(x - 5)^{2} \) to identify the operations. We see that there is a square operation applied to \( (x - 5) \). This step helps us understand the overall structure of the composite function.

Step 2: Determine the outer function \( g(x) \)

Since the outermost operation is squaring, we determine that \( g(x) \) should be the square function, so \( g(x)=x^{2} \). This is because in the composite function \( g(f(x)) \), \( g \) is the outer function that operates on the result of \( f(x) \).

Step 3: Determine the inner function \( f(x) \)

Now that we know \( g(x)=x^{2} \), the inner function \( f(x) \) should be the expression inside the square, which is \( x - 5 \). So we find that \( f(x)=x - 5 \).

Step 4: Verify the composition

To make sure we have decomposed the function correctly, we compose \( g \) and \( f \). We substitute \( f(x) \) into \( g(x) \): \( g(f(x))=g(x - 5)=(x - 5)^{2} \), which matches \( h(x) \). This verification step confirms that our decomposition is correct.

Answer:

1. Looking at \( h(x) \), notice the function is squared and see that the exponent 2 is being applied to \( (x - 5) \)
2. Find what \( g(x) \) would have to be to show a squared function. \( g(x)=x^{2} \)
3. Since \( g(x)=x^{2} \), \( f(x) \) can now be found. \( f(x)=x - 5 \)
4. Check the composition of \( f(x)=x - 5 \) and \( g(x)=x^{2} \) to verify \( h(x)=(x - 5)^{2} \). \( g(f(x))=g(x - 5)=(x - 5)^{2}=h(x) \)