Inverses of bijective functions. Whenever I see a $1-1$  and onto function from $\mathbb{R}$ to $\mathbb{R}$,I wonder how do I even go about finding the inverse.

Is there a general way of finding the inverse of a bijective function  $f :\mathbb{R}\to\mathbb{R}$? Or is it really possible that inverse of all bijective functions can be written down as a formula? 


There are an uncountable number of bijections on $\mathbb{R}$, and only a countable set of "formulas" consisting of finite strings of mathematical symbols from some (finite) alphabet. Therefore, by the pigeonhole principle, there are bijections on $\mathbb{R}$ where no formula (with any set of finite mathematical symbols) exists. Thus, there can be no general algorithm for finding inverses of bijections on $\mathbb{R}$.
