inverse of $f(x,y) = (y+2,x+2)$

Question

I'm trying to find the inverse of a function with two variables, such as $f(x,y) = (y+2,x+2)$. This is over real numbers.

It appears here that $x = y + 2$ and $y = x + 2$, so $f(1,2) = (4,3)$? And if this assumption is correct then $f^{-1}(1,2)$ would yield an ordered pair of $(3,4)$?

What would be the inverse function?

Answer 1

The inverse function is simply $f^{-1}(x,y)=(y-2,x-2)$, which you can verify with your first point.

Answer 2

Let $(z,w):= f(x,y) = (y+2,x+2)$ and solve for $x$ and $y$ in terms of $z$ and $w$.

$f^{-1}(z,w) = (x,y) = (w-2,z-2)$