Let $A$ be a set and $M$ a left $R$ -module. If $f\!: A \to M$ and $g\!: A \to M$ , then one may define the sum of functions $f$ and $g$ as the following function $f\!+\!g\!: A \to M$ : $$(f\!+\!g)(x) = f(x)\!+\!g(x) \quad \forall x \in A$$ If $r$ is any element of the ring $R$ , then the scalar-multiplied function $rf\!: A \to M$ is defined as $$(rf)(x) = r\!\cdot\!f(x) \quad \forall x \in A.$$

Let $A$ again be a set and $K$ a field or a skew field. If $f\!: A \to K$ and $g\!: A \to K$ , then one can define the product of functions $f$ and $g$ as the function $fg\!: A \to K$ as follows: $$(fg)(x) = f(x)\!\cdot\!g(x) \quad \forall x \in A$$ The quotient of functions $f$ and $g$ is the function $\displaystyle\frac{f}{g}\!: \{a\in A\,\vdots\;\; g(a) \neq 0\} \to K$ defined as $$\frac{f}{g}(x) = \frac{f(x)}{g(x)} \quad \forall x \in A\!\smallsetminus\!\{a\in A\,\vdots\;\; g(a) = 0\}.$$

In particular, the incremental quotient of functions $\frac{f(y)-f(x)}{y-x}$ , as $y$ tends to $x$ , gave rise to the important concept of derivative. As another example, we can with a clear conscience say that the tangent function is the quotient of the sine and the cosine functions.