### Instruction: If $f(x)=\frac{x^5-1}3$, find $f^{-1}(-31/96)$.### Response: To find the inverse of a function, I need to switch the roles of x and y and solve for y. So I write y = f(x) as:

\begin{align*}
y&=\frac{x^5-1}3
\end{align*}
Then I swap x and y:

\begin{align*}
x&=\frac{y^5-1}3
\end{align*}
Now I need to isolate y on one side.
I start by multiplying both sides by 3:

\begin{align*}
3x&=y^5-1
\end{align*}
Then I add 1 to both sides:

\begin{align*}
3x+1&=y^5
\end{align*}
Finally, I take the fifth root of both sides:

\begin{align*}
\sqrt[5]{3x+1}&=y
\end{align*}
This is the inverse function, $f^{-1}(x)$.