Questions: Find (f(x)) and (g(x)) such that (h(x)=(f circ g)(x)). [ h(x)=fracx^6-8x^6+8 ] Choose the correct answer below. A. (f(x)=x^6, g(x)=fracx-8x+8) B. (f(x)=fracx-8x+8, g(x)=x^6) C. (f(x)=fracx^6-xx^6+x, g(x)=8) D. (f(x)=frac1x^6+8, g(x)=x^6-8)

Solution Steps

To solve this problem, we need to express the given function \( h(x) = \frac{x^6 - 8}{x^6 + 8} \) as a composition of two functions \( f(x) \) and \( g(x) \) such that \( h(x) = (f \circ g)(x) = f(g(x)) \). We will test each option by substituting \( g(x) \) into \( f(x) \) to see if it results in \( h(x) \).

We are given the function \( h(x) = \frac{x^6 - 8}{x^6 + 8} \) and need to express it as a composition of two functions \( f(x) \) and \( g(x) \) such that \( h(x) = (f \circ g)(x) = f(g(x)) \).

We will evaluate the provided options to find which pair of functions \( (f, g) \) satisfies the equation \( h(x) = f(g(x)) \).

1. Option A: \( f(x) = x^6, g(x) = \frac{x - 8}{x + 8} \)

   • \( f(g(x)) = f\left(\frac{x - 8}{x + 8}\right) = \left(\frac{x - 8}{x + 8}\right)^6 \)

2. Option B: \( f(x) = \frac{x - 8}{x + 8}, g(x) = x^6 \)

   • \( f(g(x)) = f(x^6) = \frac{x^6 - 8}{x^6 + 8} \)

3. Option C: \( f(x) = \frac{x^6 - x}{x^6 + x}, g(x) = 8 \)

   • \( f(g(x)) = f(8) = \frac{8^6 - 8}{8^6 + 8} \)

4. Option D: \( f(x) = \frac{1}{x^6 + 8}, g(x) = x^6 - 8 \)

   • \( f(g(x)) = f(x^6 - 8) = \frac{1}{(x^6 - 8)^6 + 8} \)

Upon evaluating the options, we find that:

• Option B correctly simplifies to \( h(x) \): \[ f(g(x)) = \frac{x^6 - 8}{x^6 + 8} = h(x) \]

Final Answer