Which functions can be represented as matrices? I was reading the intuition for the associativity of matrix multiplication and it was given to be analogical to composition of functions. So which functions can be represented as matrices and how?

(Finite dimensional) Linear transforms have matrix representations. These are maps $T:V \to W$ such that $$ T(cv_1) = cT(v_1) $$ and $$ T(v_1+v_2) =T(v_1) + T(v_2) $$ for a constant $c$ and any $v_1, v_2 \in V$.

Since linear maps are special functions from one vector space to another, their compositions are associative just like compositions of any functions.