Questions: If f(x)=x^3+5 and g(x)=∛(x-5), are f(x) and g(x) inverse functions? Are f(x) and g(x) inverse functions? A. Yes, because the compositions f(g(x))=x and g(f(x)) ≠ x. B. No, because the compositions f(g(x))=x and g(f(x))=1. C. No, because the compositions f(g(x)) ≠ x and g(f(x))=x. D. Yes, because the compositions f(g(x))=x and g(f(x))=x. E. Yes, because the compositions f(g(x))=1 and g(f(x))=1.

Solution Steps

We have the functions defined as follows: \[ f(x) = x^3 + 5 \] \[ g(x) = \sqrt[3]{x - 5} \]

We need to compute the compositions \( f(g(x)) \) and \( g(f(x)) \).

1. Calculate \( f(g(x)) \): \[ f(g(x)) = f\left(\sqrt[3]{x - 5}\right) = \left(\sqrt[3]{x - 5}\right)^3 + 5 = (x - 5) + 5 = x \]

2. Calculate \( g(f(x)) \): \[ g(f(x)) = g(x^3 + 5) = \sqrt[3]{(x^3 + 5) - 5} = \sqrt[3]{x^3} = x \]

From the calculations:

• We found that \( f(g(x)) = x \).
• We also found that \( g(f(x)) = x \).

Since both compositions yield the identity function \( x \), we conclude that the functions \( f(x) \) and \( g(x) \) are indeed inverse functions.

Final Answer