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Rational Functions
Functions in the form:)()(
)( x D x N
x R =
N(x) and D(x) are POLYNOMIALS
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2
1)(
−
=
x x f
x 2.1 2.01 2.001 2.0001
2.00001
f(x) 10 100 1,000 10,000
100,000
x 1.9 1.99 1.999 1.9999
1.99999
f(x) -10 -100 -1,000 -10,000
-100,00
A Vertical
Asymptote
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56
)(−
=
x x f
x=5
y=0
I’malmost at
zeroMe
too!
A Vertical
Asymptote
HorizontalAsymptote
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+ ere are t e Asymptotes
5
6(x)a)
−=
x f
2
)5(
6g(x) b)
−
=
x
x→5 , f(x) → _____
x→∞ , f(x) → _____
x→5 , f(x) → _____
x→∞ , f(x) → _____
Vertical As m!tote"s# $ori%ontal As m!tote
&' ______________ ' ______________
Vertical As m!tote"s# $ori%ontal As m!tote
&' ______________ ' ______________
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56
)(−
=
x x f
x=5
y=0
A Vertical
Asymptote
HorizontalAsymptote
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2)5(6
)( −=
x x f
x=5
y=0
A VerticalAsymptote
HorizontalAsymptote
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oes i*ision #y zeroA.+A/ create an
Asymptote
24(x)
2
−
−=
x x f
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x 2.1 2.01 2.001 2.000
1
2.000
1f(x) '1 '01 '001 '000
1'000
1x 1.9 1.99 1.999 1.999 1.999
9f(x) 2'3 2'33 2'333 2'333 2'333
3
2
4(x)
2
−
−=
x
x f
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)2()2()2)(2(
24
(x)
2
+=−
−+=
−
−=
x x x x
x x
f
f(x) has the same (ra!h as (x+2) )ut, it*s still undefined at x=2
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Vertical As !"totes o a #ationalFunction
Let
where p( x) and q( x) have no common factors other than 1.
To locate the vertical asym totes of ! determine thereal n"mbers x where the denominator is #ero! b"t then"merator is non#ero.
( )( ) ( )
p x f x q x=
( ) f x
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.et f #e a rational unction )e(ne) #y
$ ere n is t e )egree o t e numerator an) m is t e )egree o t e )enominator' ".oo4 att e lea)ing terms'&
1' I n $ m, f as no orizontal asymptote'
5' I n %m, t en t e line y & ' "t e x -a6is& ist e orizontal asymptote o f '
2' I n & m, t en t e line is t e
1 21 2 1 '
1 21 2 1 '
...( )...
n n nn n n
m m mm m m
a x a x a x a x a f xb x b x b x b x b
− −
− −
− −
− −
+ + + + +=+ + + + +
$ori%ontal As !"totes o a #ationalFunction
n
m
a y
b=
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7ractice 7ro#lem 1
11
)(−
=
x x f
x .& .9 .99 .999f(x) -5 -10 -100 -1000
x 1.& 1.1 1.01 1.001f(x) 5 10 100 1000
Vertical As m!tote"s#
&' ______________
$ori%ontal As m!tote
' ______________
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7ractice 7ro#lem 1 - 8rap
11
)(−
=
x x f
Vertical As m!tote"s#
&' +
$ori%ontal As m!tote
'
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7ractice 7ro#lem 5
1)( 2
2
−=
x x
x f x .& .9 .99 .999
f(x) -1 -15'9 -1 :'9 -1 39
x 1.& 1.1 1.01 1.001f(x) ;' 1:'53 1;5'2 1;05
x '.& '.9 '.99 '.999
f(x) -1 -15'9 -1 :'9 -1 39
x '1.& '1.1 '1.01 '1.001
f(x) ;' 1:'53 1;5'2 1;05
Vertical As m!tote"s#
&' ______________
$ori%ontal As m!tote
' ______________
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7ractice 7ro#lem 5 - 8rap
1)( 2
2
−=
x x
x f
Vertical As m!tote"s#
&' -+, +
$ori%ontal As m!tote
' .
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7ractice 7ro#lem 2
x x
x f −
+=
55
)(
Vertical As m!tote"s#
&' ______________
$ori%ontal As m!tote
' ______________
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7ractice 7ro#lem
1)( 2 −
=
x x
x f
Vertical As m!tote"s#
&' ______________
$ori%ontal As m!tote
' ______________
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7ractice 7ro#lem ;
15
)( 22
+
−+=
x x x
x f
Vertical As m!tote"s#
&' ______________
$ori%ontal As m!tote
' ______________
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7ractice 7ro#lem <
11
)(2
+
−=
x x
x f
Vertical As m!tote"s#
&' ______________
$ori%ontal As m!tote
' ______________
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rational f"nction will have a slant (obli*"e) asym tote
if the degree of the n"merator is exactly one greater thanthe degree of the denominator.
To find an e*"ation of a slant asym tote! divide the
n"merator of the f"nction by the denominator.
The *"otient will be linear and the slant asym tote will beof the form y & *"otient.
(lant )*bli+ue, As !"totes o a #ationalFunction
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7ractice 7ro#lem again
1)( 2 −
=
x x
x f
Vertical As m!tote"s#&' -+, +
$ori%ontal As m!tote' None
Slant As m!tote
'''1' 22 −−−−− x x x x x x
)1'( 2 x x x −−−
x
y = x
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7ractice 7ro#lem :
Vertical As m!tote"s#&' -., .
$ori%ontal As m!tote' None
Slant As m!tote
'''+' 22 −−−−− x x x x x x
)2,'( 2 x x x −−−
x2,
( ) 2 + x f x
x=
−
y = 3 x
