Rational function

A function ƒ of the form

p( x )
f (x) = ,
q( x )
where p(x) and q(x) are polynomials,
with q(x) ≠ 0, is called a rational
function.

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 Rational Function

Some examples of rational functions are

1 x +1 3x 2 − 3x − 6
f (x) = , f (x) =
, and f (x)
x 2x + 5 x − 3
2
x 2 + 8 x + 16

Since any values of x such that q(x) = 0 are

excluded from the domain of a rational
function, this type of function often has a
discontinuous graph, that is, a graph that
has one or more breaks in it.
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 The Reciprocal Function

The simplest rational function with a

variable denominator is the reciprocal
function, defined by
1
f (x) = .
x

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 1
RECIPROCAL FUNCTION f ( x ) =
x
Domain: (–∞, 0) ∪ (0, ∞) Range: (–∞, 0) ∪ (0, ∞)
x y
–2 –½
–1 –1
–½ –2
0 undefined

½ 2
1 1  It is an odd function, and its graph is
2 ½ symmetric with respect to the origin.

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 Example 2 GRAPHING A RATIONAL
FUNCTION
2
Graph f ( x ) = . Give the domain and
range. x +1
Solution
2
The expression x +1 can be
 1 
written as  x + 1 indicating that
2
the graph may be obtained by
1
shifting the graph of y =
x
to the left 1 unit and stretching
it vertically by a factor of 2.

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 Example 2 GRAPHING A RATIONAL
FUNCTION
2
Graph f ( x ) = . Give the domain and
range. x +1
Solution
The horizontal shift affects the
domain, which is now
(–∞, –1) ∪ (–1, ∞) .
The line x = –1 is the vertical
asymptote, and the line y = 0
(the x-axis) remains the
horizontal asymptote. The
range is still (–∞, 0) ∪ (0, ∞).
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Asymptotes

Let p(x) and q(x) define polynomials. For the

p( x )
rational function defined by f ( x ) = , written in
q( x )
lowest terms, and for real numbers a and b:

1. If |ƒ(x)| → ∞ as x → a, then the line is a

vertical asymptote.
2. If ƒ(x) → b as |x| → ∞, then the line y = b is
a horizontal asymptote.

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Determining Asymptotes

To find the asymptotes of a rational function

defined by a rational expression in lowest
terms, use the following procedures.
1. Vertical Asymptotes
Find any vertical asymptotes by setting
the denominator equal to 0 and solving for
x. If a is a zero of the denominator, then
the line x = a is a vertical asymptote.
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Determining Asymptotes

2. Other Asymptotes
Determine any other asymptotes.
Consider three possibilities:
(a) If the numerator has lower degree
than the denominator, then there is a
horizontal asymptote y = 0 (the x-
axis).

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Determining Asymptotes
2. Other Asymptotes
Determine any other asymptotes. Consider three
possibilities:
(b) If the numerator and denominator have the
same degree, and the function is of the form
an x n +  + a0
f (x) = , where an, bn ≠ 0,
bn x +  + b0
n

then the horizontal asymptote has

an
equation y = .
bn
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Determining Asymptotes
2. Other Asymptotes
Determine any other asymptotes. Consider three
possibilities:
(c) If the numerator is of degree exactly one
more than the denominator, then there will
be an oblique (slanted) asymptote. To
find it, divide the numerator by the
denominator and disregard the remainder.
Set the rest of the quotient equal to y to
obtain the equation of the asymptote.
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 Example 4 FINDING ASYMPTOTES OF GRAPHS OF
RATIONAL FUNCTIONS
For each rational function ƒ, find all
asymptotes.
x +1
a. f ( x ) =
(2 x − 1)( x + 3)
Solution To find the vertical asymptotes, set the
denominator equal to 0 and solve.
(2 x − 1)( x + 3) =
0
=
2x − 1 0 or =
x +3 0 Zero-property
1
x= or x = −3 Solve each
2 equation.
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 Example 4 FINDING ASYMPTOTES OF GRAPHS OF
RATIONAL FUNCTIONS

The equations of the vertical asymptotes are x = ½

and x = –3.
To find the equation of the horizontal asymptote,
divide each term by the greatest power of x in the
expression. First, multiply the factors in the
denominator.
x +1 x +1
=f (x) =
(2 x − 1)( x + 3) 2 x + 5 x − 3
2

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 Example 4 FINDING ASYMPTOTES OF GRAPHS OF
RATIONAL FUNCTIONS
For each rational function ƒ, find all
asymptotes.
2x + 1
b. f ( x ) =
x −3
Solution Set the denominator x – 3 = 0 equal
to 0 to find that the vertical asymptote has
equation x = 3. To find the horizontal
asymptote, divide each term in the rational
expression by x since the greatest power of x
in the expression is 1.
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 Example 4 FINDING ASYMPTOTES OF GRAPHS OF
RATIONAL FUNCTIONS
For each rational function ƒ, find all
asymptotes.
x2 + 1
c. f ( x ) =
x −2
Solution Setting the denominator x – 2 equal to 0
shows that the vertical asymptote has equation x = 2.
If we divide by the greatest power of x as before ( in
this case), we see that there is no horizontal
asymptote because

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 Example 4 FINDING ASYMPTOTES OF GRAPHS OF
RATIONAL FUNCTIONS

We use synthetic division.

21 0 1
2 4
1 2 5

The result allows us to write the function

as
5
f (x) = x + 2 + .
x −2

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 Example 4 FINDING ASYMPTOTES OF GRAPHS OF
RATIONAL FUNCTIONS

5
For very large values of |x|,x −2 is close to
0, and the graph approaches the line
y = x + 2. This line is an oblique asymptote
(slanted, neither vertical nor horizontal) for
the graph of the function.

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Steps for Graphing Functions
A comprehensive graph of a rational function
exhibits these features:
1. all x- and y-intercepts;
2. all asymptotes: vertical, horizontal, and/or
oblique;
3. the point at which the graph intersects its
nonvertical asymptote (if there is any such
point);
4. the behavior of the function on each domain
interval determined by the vertical
asymptotes and x-intercepts.
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 Example 5 GRAPHING A RATIONAL FUNCTION
WITH THE x-AXIS AS HORIZONTAL
ASYMPTOTE
x +1
Graph f ( x ) = 2 .
2x + 5 x − 3
Solution
Step 1 Since 2x2 + 5x – 3 = (2x – 1)(x + 3),
from Example 4(a), the vertical
asymptotes have equations x = ½
and x = –3.
Step 2 Again, as shown in Example 4(a),
the horizontal asymptote is the x-
axis.
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 Example 5 GRAPHING A RATIONAL FUNCTION
WITH THE x-AXIS AS HORIZONTAL
ASYMPTOTE
x +1
Graph f ( x ) = 2 .
2x + 5 x − 3
Solution
Step 3 The y-intercept is –⅓, since

0 +1 1 The y-intercept is
f (0 ) = − . the ratio of the
2(0) + 5(0) − 3
2
3 constant terms.

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 Example 5 GRAPHING A RATIONAL FUNCTION
WITH THE x-AXIS AS HORIZONTAL
ASYMPTOTE
x +1
Graph f ( x ) = 2 .
2x + 5 x − 3
Solution
Step 4 The x-intercept is found by solving ƒ(x) = 0.
x +1
=0
2x + 5 x − 3
2
If a rational expression is equal
x +1=0 to 0, then its numerator must
equal 0.
x = −1 The x-intercept is –1.
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 Example 5 GRAPHING A RATIONAL FUNCTION
WITH THE x-AXIS AS HORIZONTAL
ASYMPTOTE
x +1
Graph f ( x ) = 2 .
2x + 5 x − 3
Solution
Step 5 To determine whether the graph
intersects its horizontal asymptote, solve
f (0) = 0. y-value of horizontal asymptote

Since the horizontal asymptote is the x-axis, the

solution of this equation was found in Step 4. The
graph intersects its horizontal asymptote at (– 1, 0).
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 Example 5 GRAPHING A RATIONAL FUNCTION
WITH THE x-AXIS AS HORIZONTAL
ASYMPTOTE
x +1
Graph f ( x ) = 2 .
2x + 5 x − 3
Solution

Step 6 Plot a point in each of the intervals

determined by the x-intercepts and vertical
asymptotes, to get an idea of how the graph
behaves in each interval.

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 Example 5 GRAPHING A RATIONAL FUNCTION
WITH THE x-AXIS AS HORIZONTAL
ASYMPTOTE
x +1
Graph f ( x ) = 2 .
2x + 5 x − 3
Solution

Step 7 Complete
the sketch.

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 Example 6 GRAPHING A RATIONAL FUNCTION
THAT DOES NOT INTERSECT ITS
HORIZONTAL ASYMPTOTE
2x + 1
Graph f ( x ) = .
x −3
Solution

Step 1 and 2 As determined in Example

4(b), the equation of the vertical
asymptote is x = 3. The horizontal
asymptote has equation y = 2.

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 Example 6 GRAPHING A RATIONAL FUNCTION
THAT DOES NOT INTERSECT ITS
HORIZONTAL ASYMPTOTE
2x + 1
Graph f ( x ) = .
x −3
Solution

Step 3 ƒ(0) = –⅓, so the y-intercept is –⅓.

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 Example 6 GRAPHING A RATIONAL FUNCTION
THAT DOES NOT INTERSECT ITS
HORIZONTAL ASYMPTOTE
2x + 1
Graph f ( x ) = .
x −3
Solution

Step 4 Solve ƒ(x) = 0 to find any x-intercepts.

2x + 1
=0
x −3
If a rational expression is equal to
2x + 1 =0 0, then its numerator must equal 0.
1
x= − x-intercept
2
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 Example 6 GRAPHING A RATIONAL FUNCTION
THAT DOES NOT INTERSECT ITS
HORIZONTAL ASYMPTOTE
2x + 1
Graph f ( x ) = .
x −3
Solution

Step 5 The graph does not intersect its

horizontal asymptote since ƒ(x) = 2 has no
solution.

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 Example 6 GRAPHING A RATIONAL FUNCTION
THAT DOES NOT INTERSECT ITS
HORIZONTAL ASYMPTOTE
2x + 1
Graph f ( x ) = .
x −3
Solution

Step 6 and 7 The

points (–4, 1), (1, –3/2),
and (6, 13/3) are on the
graph and can be used
to complete the sketch.

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Behavior of Graphs

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 GRAPHING A RATIONAL
Example 8
FUNCTION WITH AN OBLIQUE ASYMPTOTE
x +1
2
Graph f ( x ) = .
x −2
Solution In Example 4, the vertical
asymptote has equation x = 2, and the
graph has an oblique asymptote with
equation y = x + 2. Refer to the previous
discussion to determine the behavior near
the vertical asymptote x = 2.

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 GRAPHING A RATIONAL
Example 8
FUNCTION WITH AN OBLIQUE ASYMPTOTE
x +1
2
Graph f ( x ) = .
x −2
Solution The y-intercept is – ½ , and the
graph has no x-intercepts since the
numerator, x2 + 1, has no real zeros. The
graph does not intersect its oblique
asymptote because

3.5 - 32
 GRAPHING A RATIONAL
Example 8
FUNCTION WITH AN OBLIQUE ASYMPTOTE
x +1
2
Graph f ( x ) = .
x −2
Solution
x +1
2
= x+2
x −2

has no solution. Using the y-intercept,

asymptotes, the points  4,17 

2 
and 


−1, −
2
3
,

and the general behavior of the graph near
its asymptotes leads to this graph.

3.5 - 33
 GRAPHING A RATIONAL
Example 8
FUNCTION WITH AN OBLIQUE ASYMPTOTE
x +1
2
Graph f ( x ) = .
x −2
Solution

3.5 - 34