Writing a function in a part that is linearly dependent on the dependent variable

Question

Let's say I have the function $f = f(x)$,

Under what conditions am I able to write this function as follows: $f = c + h(x)x$

where $c$ is a constant, and $h(x)$ is some function depending on $x$.

Answer 1

Always, given that you can set $$ h(x)=\frac{f(x)-c}{x}, $$ (supposed $0\notin D$, where $D$ is the domain of $f$).

If $0\in D$, then necessarily $c=f(0)$.