Questions: Evaluate each of the following expressions given the function definitions for (f, g), and (h). (f(x)=sqrtx+4) (g(x)=4x-8) (h(x)=frac7x) a. (f(g(4))=) b. (g(f(72))=) c. (h(g(f(2)))=)

Solution Steps

To solve these problems, we need to evaluate the given function compositions step by step. For each part, we will substitute the inner function's result into the outer function.

a. Evaluate \( f(g(4)) \): First, calculate \( g(4) \) and then use this result to find \( f(g(4)) \).

b. Evaluate \( g(f(72)) \): First, calculate \( f(72) \) and then use this result to find \( g(f(72)) \).

c. Evaluate \( h(g(f(2))) \): First, calculate \( f(2) \), then use this result to find \( g(f(2)) \), and finally use this result to find \( h(g(f(2))) \).

To find \( f(g(4)) \), we first evaluate \( g(4) \). Using the function definition \( g(x) = 4x - 8 \), we have: \[ g(4) = 4 \times 4 - 8 = 16 - 8 = 8 \]

Next, we substitute \( g(4) = 8 \) into the function \( f(x) = \sqrt{x+4} \): \[ f(g(4)) = f(8) = \sqrt{8 + 4} = \sqrt{12} \approx 3.464 \]

To find \( g(f(72)) \), we first evaluate \( f(72) \). Using the function definition \( f(x) = \sqrt{x+4} \), we have: \[ f(72) = \sqrt{72 + 4} = \sqrt{76} \approx 8.718 \]

Next, we substitute \( f(72) \approx 8.718 \) into the function \( g(x) = 4x - 8 \): \[ g(f(72)) = g(8.718) = 4 \times 8.718 - 8 = 34.872 - 8 = 26.872 \]

To find \( h(g(f(2))) \), we first evaluate \( f(2) \). Using the function definition \( f(x) = \sqrt{x+4} \), we have: \[ f(2) = \sqrt{2 + 4} = \sqrt{6} \approx 2.449 \]

Next, we substitute \( f(2) \approx 2.449 \) into the function \( g(x) = 4x - 8 \): \[ g(f(2)) = g(2.449) = 4 \times 2.449 - 8 = 9.796 - 8 = 1.798 \]

Finally, we substitute \( g(f(2)) \approx 1.798 \) into the function \( h(x) = \frac{7}{x} \): \[ h(g(f(2))) = h(1.798) = \frac{7}{1.798} \approx 3.893 \]

Final Answer