The function \( f(x)=(x+8)^{5} \) is one-to-one. a. Find an equation for \( f^{-1}(x) \), the inverse function. b. Verify that your equation is correct by showing that \( f\left(f^{-1}(x)\right)=x \) and \( f^{-1}(f(x))=x \). a. Select the correct choice below and fill in the answer box(es) to complete your choice. (Simplify your answer. Use integers or fractions for any numbers in the expression.) A. \( f^{-1}(x)=\square \), for \( x \neq \square \) C. \( f^{-1}(x)=\square \), for \( x \geq \square \) D. \( f^{-1}(x)=\square \), for all \( x \)

To find the inverse function of \( f(x)=(x+8)^{5} \), we follow these steps: 1. Set \( y = (x+8)^{5} \). 2. Solve for \( x \) in terms of \( y \): - Take the fifth root: \( y^{1/5} = x + 8 \) - Rearranging gives \( x = y^{1/5} - 8 \) Thus, the inverse function is \( f^{-1}(x) = x^{1/5} - 8 \). Now, let’s ensure correctness by verifying the two conditions: 1. **Check** \( f(f^{-1}(x)) = x \): \[ f(f^{-1}(x)) = f(x^{1/5} - 8) = ((x^{1/5} - 8) + 8)^{5} = (x^{1/5})^{5} = x \] 2. **Check** \( f^{-1}(f(x)) = x \): \[ f^{-1}(f(x)) = f^{-1}((x+8)^{5}) = ((x+8)^{5})^{1/5} - 8 = (x+8) - 8 = x \] So, the inverse function \( f^{-1}(x) = x^{1/5} - 8 \) is indeed correct. When filling in the choices, we’d select either C or D: C. \( f^{-1}(x)=x^{1/5}-8\), for \( x \geq -8 \) D. \( f^{-1}(x)=x^{1/5}-8\), for all \( x \) Choose the one that fits best contextually, focusing on the domain adjustment.