Section 4.5
Rational Functions
Recall:Polynomial
A polynomial P (x ) is the result of adding, subtracting and multiplying powers of x. The
domain is very simply always the whole real line.
Then considering dividing powers of x, we get rational functions.
Rational Function
P( x)
where P (x ) and Q (x) are polynomials.
Q( x)
x2  2x  1
x2
1
Examples: f ( x)  2
, g ( x )  , h( x )  4
x
x 1
x  2x2  5
A rational function has the form f ( x) 
Domain of a Rational Function
The domain of a rational function is all values of x except where Q (x) =0.
Example:
x2
x2
f ( x)  2

x  1 ( x  1)( x  1)
h( x ) 
x2  2x  1
x4  2x2  5
Domain: all real numbers except x=±1.
Domain: all real numbers (the denominator is never zero)
Effect of  :
0
Whenever there is a zero is the denominator, the function either has a hole or a vertical
asymptote there.
Examples:
1
g ( x )  has a vertical
x
asymptote at x=0.
1
x
h( x)  has a hole at x=0.
has a
x 1
x
vertical asymptote at x=1. (Otherwise looks like y=1.)
f ( x) 
Generally, there is a vertical
asymptote if the zero in the
denominator doesn’t factor out,
and just a hole if it does.
x2
has a hole at x=0.
h( x ) 
x
(Otherwise looks like y=x.)
Effect of 0 :
*
Whenever there is a zero is the numerator, the function either has a zero, a hole or a
vertical asymptote there.
Review: Axes Intercepts
Recall that the x-intercepts are where the function intersects the x-axis. This occurs
wherever the function (y-value) is zero. Finding the zeros of rational functions is very
easy: you just look for the zeros in the denominator, which you can find by factoring.
Example: f ( x) 
x 2  4 ( x  2)( x  2)

The zeros are x  2 .
x2  1
x2  1
Recall that the y-intercept occurs where the function intersects the y-axis. This occurs
when x is 0, so to find the y-intercept you just compute f (x) .
Horizontal Asymptotes
If f (x) b as x±∞, then we say that the line y=b is a horizontal asymptote.