If $\mathrm{f}$ is one-to-one, find an equation for its inverse.
\[
f(x)=x^{3}-6
\]
A. $f^{-1}(x)=\sqrt[3]{x-6}$
B. $f^{-1}(x)=\sqrt[3]{x}+6$
C. $f^{-1}(x)=\sqrt[3]{x+6}$
D. The function is not one-to-one.
Answer
Comparing this with the options given, we find that the correct answer is \(\boxed{\text{C. } f^{-1}(x)=\sqrt[3]{x+6}}\).
Step 1 ：Given that the function \(f(x)=x^{3}-6\) is one-to-one, we can find its inverse by swapping \(x\) and \(y\) and solving for \(y\).
Step 2 ：So, we have \(x=y^{3}-6\).
Step 3 ：Adding 6 to both sides, we get \(x+6=y^{3}\).
Step 4 ：Taking the cube root of both sides, we get \(y=\sqrt[3]{x+6}\).
Step 5 ：So, the inverse function \(f^{-1}(x)=\sqrt[3]{x+6}\).
Step 6 ：Comparing this with the options given, we find that the correct answer is \(\boxed{\text{C. } f^{-1}(x)=\sqrt[3]{x+6}}\).
