Let $X,Y$ are two subsets of $R$, set of all real numbers. The funciton $f:X\to Y$defined by $f\left(x\right)=x^2$ for $x\in X$ is one one but not onto if 

The given function is $f:X\to Y$ and is defined as $f\left(x\right)=x^2$

Given that it is one one but not onto

Since the function is square funciton, it gives only positive values, even the domain is negative.

To make the funciton one to one, restrict the domain to the set of all positive real numbers

Since the funciton is not onto, the range is not same as codomain, so that the range is super set of of positive real numbers. 

Hence,  $\boxed{X=R^{+},Y=R}$