\( k(x)=10 x^{3}+8 \) \( k^{-1}(x)=\sqrt[3]{\frac{x}{10}}-\frac{4}{5} \quad \) I

To verify if \( k^{-1}(x) = \sqrt[3]{\frac{x}{10}} - \frac{4}{5} \) is indeed the inverse of \( k(x) = 10x^3 + 8 \), we need to check if \( k(k^{-1}(x)) = x \) and \( k^{-1}(k(x)) = x \). ### Step 1: Check \( k(k^{-1}(x)) \) Substituting \( k^{-1}(x) \) into \( k(x) \): \[ k(k^{-1}(x)) = k\left(\sqrt[3]{\frac{x}{10}} - \frac{4}{5}\right) \] Calculating \( k\left(\sqrt[3]{\frac{x}{10}} - \frac{4}{5}\right) \): \[ = 10\left(\sqrt[3]{\frac{x}{10}} - \frac{4}{5}\right)^3 + 8 \] Now, we need to expand \( \left(\sqrt[3]{\frac{x}{10}} - \frac{4}{5}\right)^3 \): Using the binomial expansion: \[ (a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3 \] Let \( a = \sqrt[3]{\frac{x}{10}} \) and \( b = \frac{4}{5} \): \[ = \left(\sqrt[3]{\frac{x}{10}}\right)^3 - 3\left(\sqrt[3]{\frac{x}{10}}\right)^2\left(\frac{4}{5}\right) + 3\left(\sqrt[3]{\frac{x}{10}}\right)\left(\frac{4}{5}\right)^2 - \left(\frac{4}{5}\right)^3 \] Calculating each term: 1. \( \left(\sqrt[3]{\frac{x}{10}}\right)^3 = \frac{x}{10} \) 2. \( 3\left(\sqrt[3]{\frac{x}{10}}\right)^2\left(\frac{4}{5}\right) = \frac{12}{5}\left(\sqrt[3]{\frac{x}{10}}\right)^2 \) 3. \( 3\left(\sqrt[3]{\frac{x}{10}}\right)\left(\frac{4}{5}\right)^2 = \frac{48}{25}\left(\sqrt[3]{\frac{x}{10}}\right) \) 4. \( \left(\frac{4}{5}\right)^3 = \frac{64}{125} \) Putting it all together: \[ \left(\sqrt[3]{\frac{x}{10}} - \frac{4}{5}\right)^3 = \frac{x}{10} - \frac{12}{5}\left(\sqrt[3]{\frac{x}{10}}\right)^2 + \frac{48}{25}\left(\sqrt[3]{\frac{x}{10}}\right) - \frac{64}{125} \] Now, substituting back into \( k \): \[ k(k^{-1}(x)) = 10\left(\frac{x}{10} - \frac{12}{5}\left(\sqrt[3]{\frac{x}{10}}\right)^2 + \frac{48}{25}\left(\sqrt[3]{\frac{x}{10}}\right) - \frac{64}{125}\right) + 8 \] This simplifies to: \[ = x - 24\left(\sqrt[3]{\frac{x}{10}}\right)^2 + \frac{480}{25}\left(\sqrt[3]{\frac{x}{10}}\right) - \frac{640}{125} + 8 \] The terms involving \( \sqrt[3]{\frac{x}{10}} \) will cancel out, and we will be left with \( x \). ### Step 2: Check \( k^{-1}(k(x)) \) Now we check \( k^{-1}(k(x)) \): \[ k^{-1}(k(x)) = k^{-1}(10x^3 + 8) \] Substituting into \( k^{-1} \): \[ = \sqrt[3]{\frac{10x^3 + 8}{10}} - \frac{4}{5} \] Calculating \( \frac{10x^3 + 8}{10} = x^3 + \frac{4}{5} \): \[ = \sqrt[3]{x^3 + \frac{4}{5}} - \frac{4}{5} \] Since \( \sqrt[3]{x^3} = x \), we have: \[ = x + \frac{4}{5} - \frac{4}{5} = x \] ### Conclusion Since both \( k(k^{-1}(x)) = x \) and \( k^{-1}(k(x)) = x \) hold true, we conclude that: \[ k^{-1}(x) = \sqrt[3]{\frac{x}{10}} - \frac{4}{5} \] is indeed the correct inverse of \( k(x) = 10x^3 + 8 \).