Say I've got some equation in two variables, like $y=a$ where a is some expression. This expression could be $x^2$ or a rational expression or whatever, you get the point. The point is that there is an equation which related the value of one variable to the value of another; it need not be a function. Can the variable in a always be solved for?
For instance: take $y=x^2 \rightarrow x=\sqrt y$, then.
A more specific statement of this question is: given some relation $R$ which relates two variables $y$ and $x$, can the inverse of $R$ always be stated in terms of both $y$ and $x$?
2 Answers
Your example will do fine, if $$ y=x^2 $$ we can solve for $x$ in terms of $y$, but not fully; only up to knowing the sign. In some senses we got lucky here, because there are "only" two inputs to solve for. Things get worse when we want to say solve for $y$ in $$ y=\sin x $$ but still ok, because we know what the solutions ought to look like, but not exactly! Indeed, we get an infinite family.
If you want to see how unpleasant finding an inverse is try to find the inverse function for $$ f(x)=xe^x $$ When you get tired feel free to plug into wolfram alpha.
The point is, you want the "relation" expressed as a function to have an inverse, i.e. be one to one and yield a unique input $x$ for each output $y$.
The answer is NO.
It is possible to find the inverse of some common relations as well a circle, $$x^2+y^2=r^2\\ \\\Rightarrow x=\pm\sqrt{r^2-y^2}$$ But you can't always find the inverse of any relations. The proof is a counterexample. $$y=\sin(x) + e^x$$ The reason is that you cannot write a single expression for $x$. Commonly it say that it cannot expressed with elementary mathematical functions.
EDIT: No every relation have inverse as i say above. But, if a relation is a one-to-one function, then the function always have inverse.
But again, is possible that that inverse will be to hard or even impossible to find with elementary mathematical functions. For example, $y=xe^x$ is a one-to-one function, but the inverse cannot be written using the elementary functions.
