Transcript Rational Functions and Their Graphs Example x Find the Domain of this Function.

Rational Functions
and Their Graphs
Example
x

7
Find the Domain of this Function. f ( x) 
x 3
Solution:
The domain of this function is the set of all real
numbers not equal to 3.
Arrow Notation
Symbol
xa
xa
x
x
is, x
Meaning
x approaches a from the right.
x approaches a from the left.
x approaches infinity; that is, x
increases without bound.
x approaches negative infinity; that
decreases without bound.
Definition of a Vertical Asymptote
The line x  a is a vertical asymptote of the graph of a function f if f(x)
increases or decreases without bound as x approaches a.
f (x)   as x  a 
f (x)   as x  a 
y
y
f
f
x
a
x=a
a
x
x=a
Thus, f (x)   or f(x)    as x approaches a from either the left or the
right.
Definition of a Vertical Asymptote
The line x  a is a vertical asymptote of the graph of a function f if f(x)
increases or decreases without bound as x approaches a.
f (x)  as x  a 
f (x)   as x  a 
y
y
x=a
x=a
x
a
f
a
x
f
Thus, f(x)   or f(x)    as x approaches a from either the left or the
right.
Locating Vertical Asymptotes
If
p( x)
f ( x) 
q( x)
is a rational function in which
p(x) and q(x) have no common factors and a is a
zero of q(x), the denominator, then x = a is a
vertical asymptote of the graph of f.
Definition of a Horizontal
Asymptote
The line y = b is a horizontal asymptote of the graph of a function f if f(x)
approaches b as x increases or decreases without bound.
y
y
y
y=b
y=b
f
f
x
f
y=b
x
x
f(x)  b as x  
f(x)  b as x  
f (x)  b as x 
Locating Horizontal Asymptotes
Let f be the rational function given by
an x n  an 1 x n1  ... a1 x  a0
f (x) 
, an  0,bm  0
m
m 1
bm x  bm1 x  ... b1 x  b0
The degree of the numerator is n. The degree
of the denominator is m.
1. If n<m, the x-axis, or y=0, is the horizontal
asymptote of the graph of f.
2. If n=m, the line y = an/bm is the horizontal
asymptote of the graph of f.
3. If n>m,t he graph of f has no horizontal
asymptote.
Strategy for Graphing a Rational Function
p( x)
f
(
x
)

Suppose that
q ( x) where p(x) and q(x) are
polynomial functions with no common factors.
1. Find any vertical asymptote(s) by solving the equation
q (x)  0.
2. Find the horizontal asymptote (if there is one) using
the rule for determining the horizontal asymptote of a
rational function.
3. Use the information obtained from the calculators
graph and sketch the graph labeling the asymptopes.
Sketch the graph of
2x  3
f ( x) 
5 x  10
2x  3
f ( x) 
5 x  10
• The vertical asymptote is x = -2
• The horizontal asymptote is y = 2/5
2x  3
f ( x) 
5 x  10
10
8
6
4
2
-10 -8 -6 -4 -2
2
-2
-4
-6
-8
-10
4
6
8 10