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?

First, note that this is not always the case.
As a geometric intuition, more than anything, consider the following:
the inverse of a function, $f^{-1}(x)$, graphically, is a reflection of $y=f(x)$in the line $y=x$ (note that a function has an inverse iff it is injective).
So, with regards to this statement, we should ask ourselves:"Under what circumstances is a function self-inverse (i.e. its own inverse)"?
The answer is: a function $f(x)=f^{-1}(x) \iff$ the graph $y=f(x)$ is symmetric about the line $y=x$, which is the case in this question, as we can see from this graph:

