### Functions: Rational functions

### The notion of rational function

The *quotient of the functions* #f# and #g# is the function #\dfrac{f}{g}# defined by #\dfrac{f}{g}(x) = \dfrac{f(x)}{g(x)}#. Such a function is defined for all real numbers #x# with #g(x)\ne0# that lie in the *domain* of #f# and in the domain of #g#.

Rational function

A **rational function** is a *function* of the form #\dfrac{p}{q}#, in which #p# and #q# are *polynomial functions*.

Such a function is defined for all real numbers #x# that are not a solution of #q(x)=0#.

If #q# is the constant function #0#, then this is a serious limitation: the rational function is not defined anywhere. But otherwise the number of points where the function #\dfrac{p}{q}# is not defined is at most the degree of #q(x)#.

Polynomial functions themselves are special rational functions that are of form #\dfrac{p}{q}#, being those in which #q(x)# is a constant (not equal to #0#).

Rules of calculation for rational functions

Let #p#, #q#, #r#, and #s# be polynomials. The following rules of calculation hold on domains where all rational functions involved are defined.

- #\displaystyle \dfrac{p}{q}+\dfrac{r}{s} = \dfrac{s\cdot p+q\cdot r}{q\cdot s}#
- #\displaystyle \dfrac{p}{q}+\dfrac{r}{q} = \dfrac{ p+ r}{ q}#
- #\displaystyle \dfrac{r\cdot p}{r\cdot q}= \dfrac{p}{q}#
- #\displaystyle r\cdot \dfrac{p}{q}= \dfrac{r\cdot p}{q}#
- #\displaystyle\dfrac{ \dfrac{p}{q}}{ \dfrac{r}{s}}= \dfrac{s\cdot p}{q\cdot r}#

These rules are the usual rules for rational numbers, whose numerator and denominator are integers. They also apply for polynomials, keeping in mind that the sets of points where the function is defined, may differ in the left hand and right hand side.

Draw the graph of #f(x)=\dfrac{x}{x^2+1}#.

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