Before you can try and find the inverse of a function you need to determine if one exists. For an inverse to exist exactly one element of the domain must map to each element of the co-domain (though you can always re-define the co-domain to only include elements that are mapped to).

So for example f(x) = x+2 (from reals to reals) has an inverse as every element y is mapped to only by y-2 but f(x) = x² (from reals to reals) doesnt because both -1 and 1 map to 1 so how would we decide which of these would be the unique result of the inverse applied to 1 (we could of course define the domain to be the positive reals in which case it would have an inverse).

When finding the inverse of a function you are really looking to see what maps to each element of the range or codomain, to find the inverse of f you are looking for the element x in the domain for a given y in the range such that f(x) = y. This basically means you are reversing the process of the function.

Finding the Inverse when the function is a formula

When a function is given by a formula what you need to try and do is apply the operations of that formula backwards. This easiest way of doing this is to let the function f(x) = y. Now you know the formula to get from x to y so substitue this in for f(x). All that now needs to be done is to rearrange this equation so that x is the subject and the resulting rexpression on the otherside is only in terms of y. This function is the inverse of f, to show this lets denote it g so we g(y) = x.

Then g(f(x)) = g(y) = x since we started by letting f(x) = y and created g such that g(y) =x.
and f(g(y)) = f(x) = y

so g is indeed the inverse of f.

Example

This is most easily demostrated through an example.
Let
Then to find the inverse of f we first write

and rearrange as follows

So the inverse of f is given by the formula so we can write

Note that we have replaced the y’s with x’s, this doesnt matter as we can put any variable we like into the function but it important to make sure that you use the same variable as the parameter of the function and in the formula that defines it.