Consider the functions $f:\mathbb{R}\rightarrow\mathbb{R}$ and $g:\mathbb{R}\rightarrow\mathbb{R}$ defined by the formulas $f(x)=x^2$ and $g(y)=y^2$ $\forall x,y \epsilon \mathbb{R}$. Is it true that $f=g$ as functions?

My thougts so far: Intuitively, yes. Since the two functions are equal at every point where they are defined and are defined on the same points, the are effectively the same function. What concerns me here is the different notation of $x$ and $y$. How does that play into the problem? Are the functions still equivalent?

Thank you.