Find a formula for the inverse of the following function, if possible.
\[
s(x)=\left(x^{3}+5\right)^{\frac{1}{5}}-1
\]
Answer
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does not have an inverse function
Answer
\(\boxed{s^{-1}(x) = \left((x+1)^{5}-5\right)^{\frac{1}{3}}}\)
Step 1 ：The function given is a composition of several basic functions. To find its inverse, we need to reverse the operations in the reverse order. The operations are: cubing, adding 5, taking the fifth root, and subtracting 1. The inverse operations are: adding 1, raising to the power of 5, subtracting 5, and taking the cube root.
Step 2 ：The inverse function should be \(s^{-1}(x) = \left((x+1)^{5}-5\right)^{\frac{1}{3}}\)
Step 3 ：\(\boxed{s^{-1}(x) = \left((x+1)^{5}-5\right)^{\frac{1}{3}}}\)
