Rational Functions

A rational function can be written in the form:

where N(x) and D(x) are polynomials and D(x) is not the zero polynomial.

In general, the domain of a rational function of x includes all real numbers except x-values that make the denominator zero

 

Example:

Example:

 

 

 

*Note: All polynomial functions are rational functions

Finding Intercepts

 

                To find the x-intercept:

                                An x-intercept is when f(x)=0

                                               Set the numerator equal to zero to find the x-intercept

                                                                     N(x)=0

                                               If  D(x)=0, then  F is undefined à Vertical Asymptote

                To find the y-intercept

                                The y-intercept is when x=0

                                                Find the y-intercept by plugging in 0 for all values of x

                                                F(0)=

Finding the Domain and Asymptotes

                Definition of Vertical and Horizontal Asymptotes:

1.       The line x=a is a vertical asymptote of the graph f if f(x)à∞ or f(x)à- ∞ as xà

a, either from the right or from the left

2.       The line y=b is a horizontal asymptote of the graph of f if f(x)àb as xà∞ or xà-∞

 

                   

To find the vertical asymptote(s), set the denominator equal to zero

 

 

                        There is a vertical asymptote at x=3

 

 

To find the horizontal asymptote(s) of the graph, you must think about its end behavior (at the extremes)

     Look at the leading terms and their coefficients: 2x and x. Do you have a big number over a little number? Do you have a little over a big? Are they the same? Now divide them:

There is a horizontal asymptote at y=2

 

Sources

Precalculus with Limits. N.p.: Houghton Mifflin, 2001. Print.