sum and product and quotient of functionsProductAndQuotientOfFunctionsSum2008-01-13 05:37:512008-02-29 08:40:34Definitionfunctionsum of functionsproduct of functionsquotient of functionsscalar-multiplied function% this is the default PlanetMath preamble. as your knowledge
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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 {\em 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 {\em 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 {\em 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 {\em 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 \PMlinkescapetext{clear} conscience say that the \PMlinkname{tangent}{TrigonometricFunction} function is the quotient of the \PMlinkname{sine}{TrigonometricFunction} and the cosine functions.