<s>[INST] How come the function and the inverse of the function are the same? What is the inverse of the function:
$$f(x)=\frac{x+2}{5x-1}$$
?
Answer:
$$f^{-1}(x)=\frac{x+2}{5x-1}$$
Can one of you explain how the inverse is the same exact thing as the original equation?
 [/INST] Let $f(x)=y$. So we have $y=\frac{x+2}{5x-1}$. Now "swap" the variables so we have 
$$x = \frac{y'+2}{5y'-1}$$
where $y' = f^{-1}(x)$ (to distinguish $y'$ from $y$). Now let's solve for $y'$.
\begin{align}
x &= \frac{y'+2}{5y'-1} \\
x(5y'-1) &= y'+2 & \text{multiply both sides by } 5y'-1 \\
5xy' - x&=y'+2 & \text{distribute $x$ on the left side} \\ 
5xy'-y'-x&=2 & \text{move all $y'$ to left side} \\
5xy'-y'&=x+2 & \text{and add $x$ to right side} \\
(5x-1)y'&=x+2 & \text{factor out $y'$ on left side} \\
y'&=\frac{x+2}{5x-1} & \text{divide both sides by $5x-1$}
\end{align}
Thus, $f^{-1}(x)=y'=\frac{x+2}{5x-1}=y=f(x)$.
</s>