A quick clarification question, what is an equivalence class of a function? For example if you have an identity function on all integers $I_{Z}$, what would $[I_{Z}]$ = ? I know that when you have a relation, the equivalence class of an element is composed of all the values its related to, but how does that look like when you have a function?
The actual question asks for a function that is not equal to that identity function but fulfills some conditions one of which is that it is part of its equivalence class, but that really confuses me, so the class is composed of functions?
EDIT: This is the question (sorry I didn't want to ask for help with my homework... it's just the concept I can't grasp): Let $I_{Z}$ be the identity function on $\Bbb{Z}$ (so $I_{Z}(x) = x$ for all $x \in Z$). Find a function $f \neq I_{Z}$ which belongs to the equivalence class $[I_{Z}]$.
If this is the same equivalence relation as in your earlier question, yes, $[f]$ is a set of functions: it’s the set of all functions $g:\Bbb Z\to\Bbb Z$ such that $g-f$ is a constant function. Note that $I_{\Bbb Z}(n)=n$ for each $n\in\Bbb Z$. Thus, $[I_{\Bbb Z}]$ is the set of all functions $g:\Bbb Z\to\Bbb Z$ such that $g(n)-n$ is a constant that does not depend on $n$. Can you think of a function $f$ such that $f(n)-n$ is always $2$, say? Or $7$? Or any other fixed integer?