How to use the inverse function theorem? I have a function $F (x, y) = (x^2+y^2, xy)$ and I need to show that it has an inverse. How do I find the inverse of this function using the inverse function theorem? I have not learned this before in all of my math classes and I just need some brushing up on how to use it. 

Thank you for your assistance.

edit: 
On the set
{(x, y) : −x &lt; y &lt; x}
For $F(x,\, y) = (x^2+y^2,\; xy)$ denote
$$\begin{cases}u = x^2+y^2, \\
v = xy \end{cases}$$
Then
$$\begin{cases}
u+2v=(x+y)^2, \\
u-2v=(x-y)^2 
\end{cases} $$
On the given set $\{(x, y):\;\; −x &lt; y &lt; x \} $ we have $$ x+y&gt;0,\;\;x-y&gt;0, \quad J_F(x,\, y)=2(x^2-y^2)\ne{0},$$ therefore $F$ is invertible and 
$$
\begin{cases}
x+y=\sqrt{u+2v}, \\
x-y=\sqrt{u-2v} 
\end{cases}$$
Thus
$$\begin{cases}
2x=\sqrt{u+2v}+\sqrt{u-2v},\\
2y=\sqrt{u+2v}-\sqrt{u-2v}.
\end{cases}$$
when you calculate jacobian matrix and found value p,where this matrix's determinant is not zero,there maybe you could you this formula. i would be happy if it helps you.also  you could check  all formula in wikipedia's site
.for single variable .let suppose that

$y=\sqrt{x}$

and   let us replace  places for x and y,or we get
$x=\sqrt{y}$

from which
$y=x^2$