As I understand it , a function $z=f(x,y)$ would be one-to-one or injective if there is only one unique $(x_1,y_1)$ pair which yields some value $f(x_1,y_1)=z_1$.
In this definition, the pair of variables $(x,y)$ is the entity that maps one-to-one to $z$.
Is there a "partial" equivalent to this that treats $x$ and $y$ separately?
I want to say something like "$z=f(x,y)$ is one-to-one with respect to $x$ for all $y$" , meaning that for any choice of $y_1$ in the domain of $y$, $f(x,y_1)$ maps $x$ to $z$ one-to-one. This property would mean that I can always "invert" a function $z=f(x,y)$ into $x=f'(z,y)$, but I can't necessarily do the same for $y=f''(x,z)$.
In general , for a multi-variable function , replace "$y$" with "all other variables."
Is there a concise terminology for this?
There is no well-known standard way to state that.
(1) I would state it in 2 ways , like this :
We have a function $y=f(x_1,x_2,x_3,\cdots,x_n)$ where there is a one-to-one correspondence between $y$ & $x_1$ , that is :
When all other variables are held constant , $y$ & $x_1$ are invertible.
Example :
$z=x(w^2+y^3+y^w)$
$x=z/(w^2+y^3+y^w)$
Here "$x$ & $z$ are invertible" & "there is a one-to-one correspondence between $x$ & $z$".
In that way , it is not necessary to limit it to variables : We can extend to expressions too , like this :
$Z=Y^W\sin(X)$
$\sin(X)=Z/Y^W$
Here "$\sin(X)$ & $Z$ are invertible" & "there is a one-to-one correspondence between $\sin(X)$ & $Z$".
(2) I have two more ways (without talking about inverse & injectivity) :
The third alternate way would be to simply state it like this :
While $z$ can be explicitly evaluated with all the other variables , $x$ too can be explicitly evaluated with all the other variables.
The fourth alternate way : I might state that when given $z=f(x,y,w)$ , $z$ is already isolated to one side , while $x$ too can be similarly isolated to one side.
These four ways are not widely used , though these are suitable to cover OP Case.
It is worth high-lighting that there is no well-known standard way to state what OP is looking for.