What is this kind of transformation of a multivariable function called and how is it denoted?

Question

Let $z = f(x, y)$ where $f(x, y)$ is not a vector-valued function. What is it called to find $x$ terms of $z$ and $y$? Here's an example of what I mean: $$z = f(x, y) = x+ y $$ Now $x$ can be expressed in terms of $z$ and $y$ with the function $g(z, y) = z-y$. What is the transformation from $f(x,y)$ to $g(z,y)$ called and how is it denoted? Obviously in this example $g(z, y) = f(z, -y)$ but this only works for addition. For instance this transformation is completely different: $$z = f(x, y) = x*y \longrightarrow x = g(z, y) = \frac{z}{y}$$

where $x = f(z, 1/y)$. This seems very similar to the inverse of a function but it's not quite the same.

Answer 1

Note that given $y$, you can define a function $h_y(x)=f(x,y)$, and then you're just looking for the inverse of $h_y$, provided it exists.

Answer 2

I think of this as a case of the $=$ sign being overloaded. In the first case it's a definition, in the second it's a computation. If we are careful about this we have $z:= x+ y$ and calculate $y=z-x$ given that definition. From this we can also see that if we define $y:=z-x$ we can calculate $z=x+y$. This implies the two definitions are logically equivalent and I would call them logically equivalent. I'm not sure anything has actually transformed however.

Answer 3

$z=f(x,y)$ represents the parametrization of a surface. $x=g(y,z)$ is a different parametrization of the same surface (assuming it exists). It's called a reparametrization.

The actual parametrization is $(x,y,f(x,y))$. The more general form is $(x(u,v), y(u,v), z(u,v))$. We just picked $u=x$ and $v=y$. If we make a different choice, like $u=y$ and $v=z$, we reparametrize.