### Functions: Fractional functions

### Quotient functions

We have already seen different types of fractional functions. *First* we looked at the function #f(x)=\tfrac{1}{x}#. *Next* we investigated power functions with a negative exponent of the form #f(x)=\tfrac{a}{x^n}#. *After that* we dealt with linear fractional functions of the form #f(x)=\tfrac{ax+b}{cx+d}#. Finally, we will look at quotient functions.

Quotient functions

A quotient function is a function of the form \[f(x)=\frac{\blue A}{\green B}\] where #\blue A# and #\green B# are polynomials.

With quotient functions the domain is equal to all values of #x# for which we have #\green B \ne 0#.

The range is dependent on the function. Any possible asymptotes can be found by considering what happens when #x# becomes very large.

#f(x)=\frac{\blue{3x+3}}{\green{x^2-4}}#

- #\frac{1}{x^2+1}# does not have vertical asymptotes
- the vertical asymptotes of #\frac{1}{x^2-1}# are #x=-1# and #x=1#

This can be seen by determining the zeroes of the denominator:

- For #\frac{1}{x^2+1}#, the denominator is equal to #0# if, and only if, #x^2+1=0#. Because the left-hand side is positive, this can never be the case. Therefore, there are no vertical asymptotes.
- For #\frac{1}{x^2-1}#, the denominator is equal to #0# if, and only if, #x^2-1=0#. Because the left-hand side can be factorized to #(x+1)\cdot(x-1)#, the values of #x# for which the denominator is #0# are #-1# and #1#. Therefore, the vertical asymptotes are #x=-1# and #x=1#.

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