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?

The inverse is not in general "the same exact thing as the original equation".
Generally, $f(x)\ne f^{-1}(x)$, but this is not always true. For example, consider the function $f(x) = -x$. This function is just the function that negates its input. Of course, if your negate your input twice, you get the original input. Put another way, to reverse the operation of negating your input, you simply negate it again.
It just happens to be the case that your function satisfies the same property. Namely,
$$f(f(x)) = \frac{f(x)+2}{5f(x)-1} = \frac{\frac{x+2}{5x-1}+2}{5\frac{x+2}{5x-1} - 1} = x$$
and hence $f(x)$ is its own inverse. (ie, to reverse $f(x)$, simply apply it again!)
