The function \( f(x)=(x+9)^{3} \) 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 \( x \neq \square \) B. \( f^{-1}(x)=\square \), for \( x \leq \square \) D. \( f^{-1}(x)=\square \), for all \( x \) b. Verify that the equation is correct. \( f\left(f^{-1}(x)\right)=f(\square) \)

To find the inverse function \( f^{-1}(x) \) of \( f(x)=(x+9)^{3} \), we start by letting \( y = f(x) \). This gives us the equation \( y = (x + 9)^{3} \). To find the inverse, we solve for \( x \) in terms of \( y \): 1. Take the cube root: \( \sqrt[3]{y} = x + 9 \). 2. Isolate \( x \): \( x = \sqrt[3]{y} - 9 \). Now swapping \( y \) for \( x \), we have: \[ f^{-1}(x) = \sqrt[3]{x} - 9. \] Thus, the correct choice is D. \( f^{-1}(x)=\sqrt[3]{x} - 9 \), for all \( x \). To verify, we need to check that \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \): 1. For \( f(f^{-1}(x)) \): \[ f(f^{-1}(x)) = f(\sqrt[3]{x} - 9) = ((\sqrt[3]{x} - 9) + 9)^3 = (\sqrt[3]{x})^3 = x. \] 2. For \( f^{-1}(f(x)) \): \[ f^{-1}(f(x)) = f^{-1}((x + 9)^3) = \sqrt[3]{(x + 9)^3} - 9 = (x + 9) - 9 = x. \] Thus, both verifications confirm that our inverse function is correct!