4.1 and 4.2
RATIONAL FUNCTIONS AND THEIR GRAPHS
Find the
Domain of
Rational
Functions
First, let’s see if we know what a rational number is. Identify the rational numbers:
3,
7,
2
,
9
7,
16 , 
3
,
4
,

,
3
2.37 , 0.333
A rational function is a function made up of the quotient of polynomial
functions. This means that rational functions can be expressed as:
f ( x) 
N x 
Dx 
where D( x)  0 .
The domain of a rational function is the set of all real numbers except the
x-values that make the denominator zero. For example, the domain of the rational
function
f ( x) 
x 2  7x  9
xx  2x  5
is the set of all real numbers except 0, 2, and -5.
Example 1
Find the domain of each rational function.
(a)
Your turn
x2  9
x3
(b)
f ( x) 
x
x 9
(c)
f ( x) 
x3
x2  9
x
x  25
(c)
f ( x) 
x5
x 2  25
2
Find the domain of each rational function.
(a)
Study
Tip
f ( x) 
f ( x) 
x 2  25
x5
(b)
f ( x) 
2
Because the domain of a rational function is the set of all real numbers except those
that make the denominator 0, you can identify such numbers by setting the denominator
equal to 0 and solving for x. Exclude the resulting values from the domain.
Arrow
Notation
The most basic rational function is the reciprocal function, defined by
f ( x) 
1
.
x
Since this is a rational function, we have to be careful with division by zero issues. So, we
can see from this equation that we’ll have to avoid x  0 since that will give division by
zero. Now, plug in some values of x and see what we get.
x
f(x)
-4
-0.25
-2
-0.5
-1
-1
-0.1
-10
-0.01 -100
0.00 Error
0.01
100
0.1
10
1
1
2
0.5
4
0.25
So, as x gets large (both positively and negatively), the function keeps the sign of x and gets
smaller and smaller. Likewise, as x approaches zero, the function keeps the same sign as x
buts start getting quite large. Here is a sketch of this graph.
7
6
5
4
3
2
1
8
6
4
2
2
4
6
8
1
2
3
4
5
6
7
What is the domain of the reciprocal function? Domain: ____________________
Observe that the graph of the reciprocal function has a break in it, and this break
makes it obvious that the graph has two branches.
Below is a summary of the arrow notation:
Arrow Notation
Symbol
Meaning
x  a
x  a
x 
x  
x approaches a from the right
x approaches a from the left
x approaches infinity (x increases without bound)
x approaches negative infinity (x decreases without bound)
Vertical
Asymptotes
A vertical asymptote is a vertical line that a graph approaches but never touches. A graph
may have many vertical asymptotes, one vertical asymptote, or no vertical asymptotes. We
use a dashed line to show asymptotes.
The line x = a is a vertical asymptote of the graph of a function f(x) if f increases or
decreases without bound as x approaches a.
Locating a
Vertical
Asymptote
If a rational function is reduced (the numerator and denominator have no common
factors), and the number a is a zero of the denominator, then the line x = a is a
vertical asymptote of the function f.
Example 2
Find the vertical asymptotes, if any, of the graph of each rational function:
(a)
Your turn
f ( x) 
x
x 9
2
(b)
f ( x) 
x3
x2  9
(c)
f ( x) 
x3
x2  9
Find the vertical asymptotes, if any, of the graph of each rational function:
(a)
f ( x) 
x
x 1
2
(b)
f ( x) 
x 1
x2 1
(c)
f ( x) 
x 1
x2 1
A Hole
In The
Graph
x2  9
. This function can be reduced as follows:
x3
x 2  9 x  3x  3
f ( x) 

 x3
x  3
x3
The original function is not defined when x  3 . However, the reduced form,  3 does
not result in a 0 denominator. Therefore, the graph does not have a vertical asymptote at
x2  9
. Notice the
x  3 . Instead, it has a hole there. Let’s look at the graph of f ( x) 
x3
location of the hole:  3,  6 .
Consider the function f ( x) 
Notice that when x  3 , the reduced form becomes f ( x)  x  3  (3)  3  6 .
Therefore, the hole appears at the point  3,  6 . The graph is a line with a hole in it!
Example 3
Locate the coordinates of the “hole” in the graph of each function.
(a)
f ( x) 
x2
x  3x  2
(b)
f ( x) 
x2  x  6
x2  x  2
(c)
f ( x) 
x 2  16
x4
Horizontal
Asymptotes
Many rational functions (but not all) have horizontal asymptotes. The line y = b is a
horizontal asymptote of the graph of f if f(x) approaches b as x increases or decreases
without bound.
A rational function may have several vertical asymptotes, but it can have at most one
horizontal asymptote.
Locating
Horizontal
Asymptotes
Example 4
Use the following to help locate horizontal asymptotes:
2.
If the degree of the numerator is equal to the degree of the denominator,
then the horizontal asymptote is found by dividing the leading term
coefficient of the numerator by the leading term coefficient of the
denominator.
3.
If the degree of the numerator is greater than the degree of the denominator,
then the function has no horizontal asymptote.
4x
f ( x)  2
2x  1
(b)
4x 2
f ( x)  2
2x 1
(c)
4x 3
f ( x)  2
2x 1
Find the horizontal asymptote, if any, of the following functions:
(a)
Graphing
Rational
Functions
If the degree of the numerator is less than the degree of the denominator,
then the x-axis (or the line y = 0) is the horizontal asymptote.
Find the horizontal asymptote, if any, of the following functions:
(a)
Your turn
1.
f ( x) 
9x 2
3x 2  1
(b)
9x
f ( x)  2
3x  1
(c)
f ( x) 
9x 3
3x 2  1
We will look at strategies for graphing rational functions.
Strategies for graphing rational functions:
Suppose f ( x) 
N x 
where N x  and Dx are polynomials with D( x)  0 .
Dx 
1. Simplify the fraction, if possible. (If you can simplify, the graph will have a hole.)
2. Find and plot the y-intercept (if there is one) by making x  0 .
3. Find the zeros of the numerator (if there are any) by solving the equation N ( x )  0 .
Then plot the corresponding x-intercepts.
4. Find the zeros of the denominator (if there are any) by solving the equation
D( x)  0 . Then sketch the corresponding vertical asymptotes.
5. Find and sketch the horizontal asymptote (if there is one) using the procedures for
finding horizontal asymptotes.
6. Plot at least one point between and one point beyond each x-intercept and vertical
asymptote.
7. Use smooth curves to complete the graph between and beyond the vertical
asymptotes.
Example 5
Graph g ( x) 
3
x2
Example 7
Graph f ( x) 
x
2
x x2
Example 6
Example 8
Graph f ( x) 
Graph f ( x) 
2x  1
x
x2  9
x 2  2x  3
Slant
Asymptotes
The graph of a rational function has a slant (or oblique) asymptote if the degree of the
numerator is one more than the degree of the denominator.
x2 1
Look below at the graph of f ( x) 
.
x 1
10
8
f(x)=
6
x2+1
x-1
4
2
-15
-10
-5
5
10
15
-2
-4
-6
-8
-10
Notice that the degree of the numerator is 2 and the degree of the denominator is 1. Since
the degree of the numerator is one more than the degree of the denominator, the function
has a slant asymptote. The asymptote is the line y  x  1 .
Finding
Slant
Asymptotes
To see that the slant asymptote is y  x  1 , observe what happens when we use
synthetic division to divide the numerator by the denominator:
x2 1
2
 x 1
.
x 1
x 1
If we ignore the remainder, the result is the equation of the slant asymptote.
If the degree of the numerator is one more than the degree of the denominator,
N x 
remainder
 mx  b 
and if
, then y  mx  b is the equation of the slant
D x 
D x 
asymptote.
Example 9
Find the slant asymptote of f ( x) 
x 2  4x  5
. Then graph the function.
x3
14
12
10
8
6
x2-4x-5
f(x)=
x-3
-15
-10
4
2
-5
5
10
15
-2
-4
-6
-8
-10
-12
Your turn
Find the slant asymptote of f ( x) 
2 x 2  5x  7
. Then graph the function.
x2
30
25
20
15
10
f(x) =
-20
-15
-10
2x2-5x+7
x-2
5
-5
5
-5
-10
-15
-20
-25
10
15
20
Example 10 Find the slant asymptote of f ( x) 
x2  x
. Then graph the function.
x 1
x2  x  2
Example 11 Find the slant asymptote of f ( x) 
. Then graph the function.
x 1
Example 12 Sketch the graph of f ( x) 
3x  4
.
x  x6
2
Example 13 An Application
A company producing electronic components used in television sets has established that
on the average, a new employee can assemble N (t ) components per day after t days of
on-the-job training, given by
N t  
50t
, t  0.
t4
Sketch the graph of N, including any horizontal and vertical asymptotes. Explain the
meaning of the graph as t increases without bound.
120
110
100
90
80
70
60
y = 50
50
40
N(t) =
30
t = -4
50t
t+4
20
10
-30
-20
-10
10
20
30
-10
-20
-30
Example 14
Function:
4x  5
8 x  16
4 x 2  10 x  9
y
2x  3
3
x  3x 2  4 x  12
y
x 2  3x  10
y
Vertical
Asymptote:
x2
x
3
2
x  5
Horizontal
Asymptote:
1
y
2
Slant
Asymptote:
Hole:
None
None
None
y  2x  2
None
None
y  x6
4

 2,  
7

Extra Practice – Graphing Rational Funcions
1 – 6. Find the domain of each rational function.
5x
f x  
1.
x4
x7
f x   2
3.
x  49
x
f x   2
5.
x  2x  3
2.
4.
6.
3x 2
x  5x  4
x7
f x   2
x  49
x8
f x   2
x  64
f x  
7 – 14. Find the vertical asymptote (if any) of each rational function.
7.
9.
11.
13.
f x  
x
x4
x3
f x  
x x  4 
x
f x  
x x  4 
x
f x   2
x 4
8.
10.
12.
14.
f x  
5x
x3
x3
f x  
xx  3
x
f x  
xx  3
x
f x   2
x 3
15 – 22. Find the horizontal (if any) of each rational function.
15.
17.
19.
21.
f x  
12 x
3x 2  1
12 x 2
f x   2
3x  1
12 x 3
f x   2
3x  1
 2x  1
f x  
3x  5
16.
18.
20.
22.
f x  
15 x
3x 2  1
15 x 2
f x   2
3x  1
15 x 3
f x   2
3x  1
 3x  7
f x  
5x  2
23 – 28. Find the slant asymptote (if any) of each rational function.
x2  1
x2  4
f  x 
23.
24.
f  x 
x2
x 1
2
2
x 1
x 4
f x  
f x  
25.
26.
x2
x 1
2
x  x6
x2  x 1
f x  
f x  
27.
28.
x3
x 1
29 – 38. Graph each rational function. Give the domain, label all asymptotes and intercepts, and
determine if the graph has any “holes” in it and label the “hole.” Finally, determine if the graph crosses its
horizontal asymptote. If it does, label the point in which that occurs.
29.
f x  
31.
f x  
33.
f x  
35.
f x  
37.
f x  
4x
x2
2x 2
x2 1
x2
2
x  x6
x2 1
x
2
x  x6
x3
Rational Functions Answer Key – Extra Practice
1.
x4
3.
x  7
x  3, x  1
5.
7.
x  4
9.
x  0 and x  4
11.
x  4 (there is a hole when x  0 )
13.
No vertical asymptote
y0
15.
y4
17.
19.
No horizontal asymptote
2
y
21.
3
yx
23.
y  x 1
25.
y  x4
27.
30.
32.
34.
36.
38.
2.
4.
6.
8.
10.
12.
14.
16.
18.
20.
22.
24.
26.
28.
f x  
2x
x 4
2
f x   2
x x2
x 2  x  12
f x  
x2  4
x2 1
f x  
x 1
x
f x  
x  32
2
x  5, x  4
All real numbers
All real numbers
x3
x  0 and x  3
x  3 (there is a hole when x  0 )
No vertical asymptote
y0
y5
No horizontal asymptote
3
y
5
yx
y  x2
yx
29.
f x  
4x
x2
14
12
10
8
6
4
2
20
15
10
5
5
10
15
2
4
6
8
10
12
14
30.
f x  
2x
x 4
2
8
6
4
2
10
5
5
2
4
6
8
10
20
f x  
31.
2x 2
x2 1
5
4
3
2
1
4
2
2
4
1
2
f x  
32.
2
x x2
2
4
3
2
1
6
4
2
2
1
2
3
4
4
6
f x  
33.
x2
x  x6
2
4
3
2
1
6
4
2
2
4
1
2
3
f x  
34.
x 2  x  12
x2  4
9
8
7
6
5
4
3
2
1
10
5
5
1
2
3
4
5
10
A close-up of the graph for 3  x  12.5 :
The graph crosses its horizontal
asymptote at the point (8, 1)
3
4
f x  
35.
10
5
6
7
8
9
10
11
12
x2 1
x
5
5
10
f x  
36.
x2 1
x 1
10
8
6
4
2
10
5
5
10
2
4
6
f x  
37.
x2  x  6
x3
18
16
14
12
10
8
6
4
2
20
15
10
5
5
2
4
6
8
10
10
15
20
38.
f x  
x
x  32
3.5
3
2.5
2
1.5
1
0.5
2
2
0.5
The graph crosses its horizontal asymptote at (0, 0) .
4
6
8