The function \( f(x)=(x+10)^{3} \) 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 \( x \leq \square \) B. \( f^{-1}(x)=\square \), for \( x \geq \square \) C. \( f^{-1}(x)=\square \), for all \( x \) D. \( f^{-1}(x)=\square \), for \( x \neq \square \)

To find the inverse function \( f^{-1}(x) \) for \( f(x) = (x + 10)^3 \), we first swap \( x \) and \( y \) and solve for \( y \): 1. Replace \( f(x) \) with \( y \): \[ y = (x + 10)^3 \] 2. Swap \( x \) and \( y \): \[ x = (y + 10)^3 \] 3. Take the cube root of both sides: \[ y + 10 = \sqrt[3]{x} \] 4. Isolate \( y \): \[ y = \sqrt[3]{x} - 10 \] Thus, the inverse function is: \[ f^{-1}(x) = \sqrt[3]{x} - 10 \] So, the correct choice is B: \[ f^{-1}(x) = \sqrt[3]{x} - 10, \text{ for } x \geq -10 \] Now, let’s verify: 1. Check \( f(f^{-1}(x)) \): \[ f(f^{-1}(x)) = f(\sqrt[3]{x} - 10) = ((\sqrt[3]{x} - 10) + 10)^3 = (\sqrt[3]{x})^3 = x \] 2. Check \( f^{-1}(f(x)) \): \[ f^{-1}(f(x)) = f^{-1}((x + 10)^3) = \sqrt[3]{(x + 10)^3} - 10 = (x + 10) - 10 = x \] Both conditions are satisfied, so the equation is correct!