What is the term for equation in $y=f(x,y)$

Question

What is the mathematical terminology for a mathematical relation which can't be expressed as $y=f(x)$ format. Rather you can say it is $y=f(x,y)$ formation. I apologize if I could not explaining it right due to my lack of mathematical knowledge.I stumbled on this equation : $$ y=1-exp(-k(\alpha + y\beta + x ) ) $$ where $\alpha,\beta, k$ are constant value. You can see that both side have $y$ . I do not know any straight forward method to solve this kind of equation, so I solve it in MATLAB using iterative approach.

What kind of equation is this (like we have linear, polynomial etc. ) ?

Sidenote: If you can provide any tutorial link where I can learn to solve this kind of equation without using any software.

Answer 1

I think "implicit function" is the term you are looking for.

Answer 2

This is a special case for a Functional Equation.

There is no general algorithm to solve this kind of equations and sometimes you need to impose assumptions on $f$ (e.g. continuity, differentiability, etc.) to solve it.

Answer 3

I would call that an "exponential equation".

Added: I would NOT call it a "functional equation" because all "unknowns" are numbers, not functions.