What is the Inverse of a 2D Function?

[n0ya]

I have a 2D function f:

f(x,y) = a + bx + cy + dxy

what is the inverse of this function?

 

[micromass]

Hi n0ya!

The function you mention will never have an inverse.

For example, if f(x,y)=2+x, then f cannot have an inverse since f(0,0)=2=f(0,1). Thus (0,0) and (0,1) are both being sent to 2. But then the inverse needs to send 2 to both (0,0) and (0,1), but this is impossible for a function.

In general, your function is one $f:\mathbb{R}^2\rightarrow \mathbb{R}$, it can have no (continuous) inverse since otherwise the plane would be homeomorphic to the line. And this cannot be.


If you had a function $f:\mathbb{R}^2\rightarrow \mathbb{R}^2$ then you might have a continuous inverse. But even then this depends of the function f...

 

[n0ya]

Thanks!