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Properties of Rational Functions
2
Learning Objectives
1. Find the domain of a rational function2. Find the vertical asymptotes of a rational function3. Find the horizontal or oblique asymptotes of a rational function
Rational Function
where and are polynomial functionsp x q x
1 0
1 0
nn
dd
p x a x a x a
q x b x b x b
1 0
1 0
, 0n
nd
d
p x a x a x aR x q x
q x b x b x b
The degree of is
The degree of is
p x n
q x d
If then is an improper rational functionn d f x
To Find the Domain
Given p x
R xq x
Solve 0q x
DomainThe domain of a rational function is all real values except where the denominator, q(x) = 0
Example
2 6 9
2
x x
x
Domain: all reals except 2x
Example
2 1
1
x
x
Domain: all reals except 1x
Example
2
2
1
3 2
x
x x
1 1
1 2
x x
x x
Domain: all reals except
1 and 2x x
Points Not in The DomainPoints not in the domains of a rational function
are either holes or vertical asymptotes
A point is a vertical asymptote if it cannot not
be compeletly factored out
If after factoring, then
the zeroes of 0 are vertical asymptotes (VA)
p x p xR x
q x q x
q x
the other zeroes of 0 are then holesq x
Holes and Vertical Asymptotes
3 is a holex
2
2
5 6
4 3
x x
x x
1 is a VAx
2 3
1 3
x x
x x
2
1
x
x
3, 1D f
x
xxR
3
5 2
3 is a VAx
Examples
Find holes and vertical asymptotes
3
2 3
xH x
x x
3 is a hole and 2 is a VAx x
Examples
Find holes and vertical asymptotes
1
1 4
xF x
x x
4 and 1 are VAx x
Example Find holes and vertical asymptotes
2
2
3 2
4
x xf x
x
1 2
2 2
x x
x x
1
2
x
x
2
2
3 2
4
x xf x
x
2 is a VAx 2 is a holex
Example
14
More on HolesPoints not in the domains of a rational function
are either holes or vertical asymptotes
A point is a vertical asymptote if it cannot not
be compeletly factored out
If after factoring, then
the zeroes of 0 are vertical asymptotes (VA)
p x p xR x
q x q x
q x
the other zeroes of 0 are then holesq x
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Holes and Vertical Asymptotes
Holes and vertical asymptotes are discontinuities, but they are very different
vertical asymptotes are non-removable discontinuities
but holes are removable discontinuities, and by the addition of a point, we can create a function continuous at that point
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Example Hole at x = 2
Holes do not appear on the graph, but are clearly indicated on the table
X-2 evenly divides both the numerator and the denominator
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Example Vertical Asymptote at x = 2
Holes do appear on the graph and are clearly indicated on the table
Example 2
1f x
x
x 0- f(x) ∞
x 0+ f(x) ∞
x 0 f(x) ∞
horizontal asymptotes (HA)
oblique (slant)
asymptotes (OA)
Horizontal and Oblique Asymptotes
End Behavior
1 0
1 0
nn
dd
p x a x a x aR x
q x b x b x b
1 There is an oblique (slant) asymptoten d
is the horizontal asymptoten
d
an d y
b
0 is the horizontal asymptoten d y
A function will not have both an oblique and a horizontal asymptote
A horizontal line is an asymptote only to the far left and the far right of the graph.
"Far" left or "far" right is defined as anything past the vertical asymptotes or x-intercepts.
Horizontal asymptotes are not asymptotic in the middle.
It is okay to cross a horizontal asymptote in the middle.
Horizontal Asymptote
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Example 2
2 5
4 3
x
x x
n d 0 is HAy
Find equation for horizontal asymptote
23
2 is HA
3y
Example
2
2
2 3 1
3 2
x x
x x
n d
Find equation for horizontal asymptote
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Example 2
2
2 3 1
3 2
x x
x x
2
2
2 3 1 2
3 2 3
x x
x x
2 23 2 3 1 2 3 2x x x x
2 26 9 3 6 4x x x x
9 3 4x x 3
5x
Find x-value(s) where f(x) crosses horizontal asymptote
25
Example
3
2
3
4
x x
x x
1n d There is a OA, no HA
Find equation for horizontal asymptote
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Example
4
2
2 1
3
x x
x
1n d There is no OA or HA
Find equation for horizontal asymptote
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Find equation for horizontal asymptote
2
2
3 2Find HA
4
x xf x
x
11
1n d y
Example
28
2
2
3 2Find HA
4
x xf x
x
11
1n d y
Example
29
Example 1 2
1 2
Resistance of a parallel circuit is
found by T
R RR
R R
1If 10 find the horizontal asymptoteR
2
2
10
10T
RR
R
10Note form is y
10
x
x
1010
1n d y 10 HorizR
This means for very large values of R2 the total resistance approaches 10 ohms.
30
Oblique Asymptotes
When the degree of the numerator is exactly one more than the degree of the denominator, the graph of the rational function will have an oblique asymptote.
Another name for an oblique asymptote is a slant asymptote.
To find the equation of the oblique asymptote, perform long division (synthetic if it will work) by dividing the denominator into the numerator and discarding the remainder
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Example Oblique Asymptote y = x+2
Y2 is the end behavior of y1
X-2 divides the numerator with a remainder
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Finding Oblique Asymptotes
1 0
1 0
If after factoring,
and 1
nn
dd
p x p xa x a x aR x
b x b x b q x q x
n d
Divide by and obtain
where is a proper rational function
r xp x q x f x
q x
f x
is the oblique asymptotey f x
or divide * by * and obtain *
where is a proper rational function
r xp x q x f x
q x
f x
33
Example
3
2
3
4
x x
x x
2 3 2
3 2
2
2
44 0 3
4
4 3
4 16
19
xx x x x x
x x
x x
x x
x
3
2 2
3 194
4 4
x x xx
x x x x
194
4x
x
Discard the remainder
and the OA is 4y x
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Example
2 3
4
x x
x x
2
2
44 0 3
4
4 3
4 16
19
xx x x
x x
x
x
2 3 194
4 4
xx
x x
194
4x
x
Discard the remainder
and the OA is 4y x
3
2
3
4
x x
x x
35
Example
3
2
3
4
x x
x x
36
Example
23 2Find OA
1
x
x
2
2
3 31 3 0 2
3 3
3 2
3 3
1
xx x x
x x
x
x
23 2 13 3
1 1
xx
x x
oblique asymptote
3 3y x
37
Example
23 2Find OA
1
x
x
1 3 0 2
3 3
3 3 5
Using synthetic division
3 3 1x R
oblique asymptote
3 3y x
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Our rational function
Our rational function and OA
Example
23 2Find OA
1
x
x
39
Example
2 3 2Find OA
2
x x
x
2
2
52 3 2
2
5 2
5 10
12
xx x x
x x
x
x
OA 5y x
40
Example
2 3 2Find OA
2
x x
x
2 1 3 2
2 10
1 5 12
Using synthetic division
5 R=12x
OA 5y x
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Our rational function
Our rational function and OA
Example
2 3 2Find OA
2
x x
x
