The function \( f(x)=x^{3}+10 \) is one-to-one. a. Find an equation for \( f^{-1} \), 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 \( x \geq \square \) B. \( f^{-1}(x)=\square \), for \( x \neq \square \) C. \( f^{-1}(x)=\square \), for \( x \leq \square \) b. Verify that the equation is correct. \( f^{-1}(x)=\square \), for all \( x \) \( f\left(f^{-1}(x)\right)=f(\square) \)

To find the inverse of the function \( f(x) = x^3 + 10 \), we can start by setting \( y = f(x) \). Thus, we rewrite it as \( y = x^3 + 10 \). Now, solving for \( x \) gives us \( x^3 = y - 10 \) and \( x = \sqrt[3]{y - 10} \). Therefore, the inverse function can be expressed as \( f^{-1}(x) = \sqrt[3]{x - 10} \), which is valid for all \( x \) since there's no restriction. Now, let's verify the equation. We will first check \( f(f^{-1}(x)) \). 1. Calculate \( f(f^{-1}(x)) \): \[ f(f^{-1}(x)) = f\left(\sqrt[3]{x - 10}\right) = \left(\sqrt[3]{x - 10}\right)^3 + 10 = (x - 10) + 10 = x \] 2. Now check \( f^{-1}(f(x)) \): \[ f^{-1}(f(x)) = f^{-1}(x^3 + 10) = \sqrt[3]{(x^3 + 10) - 10} = \sqrt[3]{x^3} = x \] Both verifications show that the original functions yield the correct results, confirming that our inverse function is indeed correct. So, summarizing the filled responses: a. \( f^{-1}(x) = \sqrt[3]{x - 10} \), for all \( x \). b. \( f^{-1}(x) = \sqrt[3]{x - 10} \), for all \( x \); \( f(f^{-1}(x)) = x \).