Introduction to Rational Functions

A Rational function is any function that looks like

To find the vertical asymptotes, factor the denominator to find the possible asymptotes

Use the zero product property to identify possible x-values for vertical asymptotes

Graph the function to find the vertical asymptotes

 $f\left(x\right)=\frac{3x-2}{x^2+7x+12}$  
Set denominator equal to zero
 $\ x^2+7x+12=0$  
 $\left(x+4\right)\left(x+3\right)=0$  
Possible asymptotes $x=-3,\ x=-4$  
Graph to find asymptotes

First, identify if the degree of the nominator is less than the degree of the denominator. If so, there is one asymptote and it is at:

 $f\left(x\right)=\frac{x^2+1}{x+2}$  

Second, if the degree of the nominator is greater than the degree of the denominator there is no horizontal asymptote. Example: The nominator increasing faster than the denominator  so there are no horizonal asymptotes.

 $f\left(x\right)=\frac{2x^2+x+1}{x^2-1}$  

The degrees of the variable in the nominator and the denominator are the same, dividing the denominator into the numerator we find that the ratio of the leading coefficients is the horizontal asymptote.