Download 2.6 Rational Functions and Asymptotes

2.6 Rational Functions and
Asymptotes
Rational Function
Rational function can be written in the form
N ( x)
f ( x) 
D( x)
where N(x) and D(x) are polynomials and D(x) is
not 0
Domain of a rational function is all real numbers
except the x-values that make D(x) = 0
Finding Domain/Range
Find the domain and the range of the function
1
f ( x) 
x
x
-1
-.5
-.1
-.001
-.0001
f(x)
-1
-2
-10
-100
-1000
x
.001
.01
.1
.5
1
f(x)
1000
100
10
2
1
Horizontal and Vertical Asymptotes
1) The line x = a is a vertical asymptotes of the
graph f if f(x)  ∞ or f(x)  -∞ as x  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)  b as x  ∞ or x  -∞
Asymptotes of a Rational Function
Let f be the rational function
where N(x) and D(x) have no common factors.
To Find A Vertical Asymptote:
The graph has a vertical asymptotes at the
zeros of D(x)
Asymptotes of a Rational Function
Let f be the rational function
where N(x) and D(x) have no common factors.
To Find A Horizontal Asymptote:
The graph of f has at most one horizontal
asymptote determined by comparing the
degrees of N(x) and D(x)
n = the degree of N(x) and d = the degree of D(x)
1. If n < d, then y = 0
2. If n = d, then y = an /bn (leading coefficients)
3. If n > d, then there is no horizontal asymptote
Finding the Asymptotes
1)
2x
f ( x)  2
3x  1
2)
2x2
f ( x)  2
x 1
Finding Horizontal and Vertical
Asymptotes
x  x2
f ( x)  2
x  x6
2
Find the a) domain, b) vertical asymptotes, and
c) horizontal asymptotes
3x 3  7 x 2  2
f ( x) 
 4 x3  5
Two Horizontal Asymptotes
A function that is not rational can have two horizontal
asymptotes. One to the right and one to the left.
x  10
f ( x) 
x 2
Word Problems
A utility company burns coal to generate
electricity. The cost C (in $) of removing p% of
the smokestack pollutants is given by
C=80,000/(100-p) for 0 < p < 100. Use your
calculator to graph the function. You are a
member of a state legislature that is considering
a law that would require utility companies to
remove 90% of the pollutants from their
smokestack emissions. The current law requires
85% removal. How much additional cost would
there be to the utility company because of the
new law?