How do we emphasize that $\displaystyle x\mapsto\frac{1}{f(x)-y}$ "makes sense" if we know $y\notin\text{im }f$?

Question

Please take a look at the following function $$x\mapsto\frac{1}{f(x)-y}$$ where $f$ is "some other function". Suppose we know $y\notin\text{im }f$, i.e. the expression in the denominator "makes sense". How do I call this function if I want to emphasize this fact?

Remark: At first, I thought about the term "well-defined". But actually it's already overloaded with other meanings.

Answer 1

You can just say it's well-defined. When you say a function is 'well-defined' all you're saying is that the thing you're defining really is a function, and not some other mathematical object (or something nonsensical).

Answer 2

Well-defined is fine imho. If $y\in im f$ then for some $x$ it would hold $f(x)=y$, and then $\frac1{f(x)-y}=\frac10$ is not (well)-defined.