Find a formula for the inverse of the following function, if possible.
\[
s(x)=\sqrt[3]{5 x+1}
\]
Answer
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\[
s^{-1}(x)=
\]
does not have an inverse function

Step 1 ：The function given is a cubic root function, which is a one-to-one function. This means that it does have an inverse. To find the inverse of a function, we need to switch the x and y (or in this case, s(x)) values and solve for y.

Step 2 ：In this case, we have \(s(x) = (5x + 1)^{1/3}\). If we switch x and s(x), we get \(x = (5s(x) + 1)^{1/3}\). Now we just need to solve this equation for s(x) to find the inverse function.

Step 3 ：We start with \(x = (5s(x) + 1)^{1/3}\). To isolate s(x), we can cube both sides to get rid of the cubic root, which gives \(x^3 = 5s(x) + 1\). Then, we can subtract 1 from both sides and divide by 5 to solve for s(x).

Step 4 ：Final Answer: The inverse function of s(x) is \(\boxed{s^{-1}(x)=\frac{x^3-1}{5}}\)