I would like to find the inverse of $X$

Question

Let $X(u,v)=(v−u,u^2−v^2,u+v)$, $(u,v)\in U=\mathbb R^2$ and $S=X(U)$.

what is the inverse of $X$

where $X$ is the function which maps a $2$D object into a $3$D object

$X^{-1}(x,y,z)$

Answer 1

Let $X^{-1}(x,y,z)=(f,g)$. Then, $$\tag1x=g-f$$ $$\tag2y=f^2-g^2$$ $$\tag3z=f+g$$ Adding $(1)$ and $(3)$, $$g=\frac{x+z}2$$ And $$f=\frac{z-x}2$$

$$X^{-1}(x,y,z)=\left(\frac{z-x}2,\frac{x+z}2\right)$$ Note that $(x,y,z)\in S$,i.e., the values of $x,y,z$ are not independent.

Answer 2

Hint: to find the inverse means that for a given point $A(a_1,a_2,a_3)\in X(U)\subset \mathbb R^3$ you have to determine what could be $(u,v)\in\mathbb R^2$ so that $X(u,v)=(a_1,a_2,a_3)$, i.e to solve the nonlinear system

$$\begin{array}{|l} &v-u=a_1\\ &u^2-v^2=a_2\\ &u+v=a_3 \end{array} $$

Note that, not all points in $\mathbb R^3$ are in the image of $X$. For example a necessary condition for $A\in X(U)$ is that $a_1a_3=-a_2$. This follows by multiplying the first and the third equation and then comparing with the second.