Let $A = \{(x,y): 0 < y < x\}$ and $f: A \to \mathbb{R}^{2}$. Let $f(x,y) = (x+y,xy)$ for $(x,y) \in A$.Find a formula for the inverse.
I'm a bit at a loss of how to even start. A simple way for 1D functions is to just swap x and y and solve for y, but I've never learned how to do this for multivariable functions
Just swap now both variables (as was also commented): The trick is to introduce two more letters: $$(u,v):=f(x,y)=(x+y,xy)$$ That is, $u=x+y,\ v=xy$. Then the quadratic $z^2-uz+v=0$ has solutions exactly $\{z_1,z_2\}=\{x,y\}$, so they can be expressed as $\displaystyle\frac{u\pm\sqrt{u^2-4v} }2$. Now $x>y$ determines that $x=\displaystyle\frac{u+\sqrt{u^2-4v} }2$ and $y$ the smaller one, and we also need to have $u^2-4v>0$, and $u,v>0$.