There are a couple of ways in which an attempt to find a formula for 
from the formula for f could fail.
First, if f is not 1-1 then the method should fail! Here's what could go
wrong. Consider 
(a parabola) with domain R and range 
. This function is not
1-1 on its domain so it should not
have an inverse. If we try to find we are stopped at some point
because of this. Try it.
(1) 
(2) 
(3) 
Well, we solved for y but the solution was not unique. As you can see, we get two y values for every positive x value and that's not a function. Each x in the domain must be associated with exactly one y value in the codomain.
A common way to get around this problem is to simply restrict the domain so
that the resulting function is 1-1. If we consider with domain 
(the right half of t
he parabola) and range then we do have a 1-1 function. Now we can find an inverse.
(1)
(2)
(3) 
So we get 
with domain and range .
Second, even when an inverse exists you might not be able to do the algebra
required to find a formula for it. Consider 
. This function
is 1-1 on R but finding requires solving the equation 
and you are not going to do that.
Actually, the ability to solve an equation algebraically is sort an
ambiguous notion if you look at it this way. The function 
with domain 
is 1-1 and has an inverse. Can you
solve 
to find it. Well, if you consider making up a new symbol 
in order to write
down 
as ''solving'' the
equation, then yes you can solve it. If you think about some other symbols
you have seen, like 
or 
, you will see that these are just
bits of notation that are used to ''solve'' for an inverse function ( 
being an inverse for 
from
above by making up a name, oh say Fred, and claiming that y=Fred(x) is
the inverse of . But let's stop there.
