vw - davis.k12.ut.us

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Unit4
H Seconddry 3
Unit 4: Rational Functions
•
4.1 -Simplifying, Multiplying and Dividing Rational Expressions
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In the previous unit we studied polynomial functions. In this unit we will expand and learn about rational
functions.
A rational function is defined as follows:
R(x) =
~~=~,
Where N(x) and D(x) are polynomial functions and D(x) '# 0
***The Domain of a rational function is all real numbers except x-values that make the denominator
zero***
~
We have already seen some specific types of rational function. Linear, quadratic, cubic, and higher order
polynomial functions are types of rational functions.
-.
Example 1: Determine whether each function is a rational function or is not a rational function. If it is not
rational, explain why.
a.
h(x)
= x2 - 6x -
1
b.
rex)=
l
lhl
Gr
2
c. g(x)
[/1
1
d. p (x)= x2+2
x +1
~
\~
-Y\\)\
= xi -
~1
vw\ (}. ~\~
~()\~
Example 2: Determine the restriction(s) ofthe domain for each variable ofthe expression, then simplify.
3xb1.,
a.-
27xb..
b. _2:,_
3x-15
;;>\. t-C ')
c.
x -5
.l(l=2-X
s )
d.
x 2 - 7x+12
x 2 + 3x - 18
( 'f --t.f)( '1.--
?J)
(y:-t~) ()(-~)
-2()(-c,)
83
Example 3: Find the domain, then simplify.
a.
2X2-!f
x-S
c.
b.-x-2
10
~)'
(-
<J)
\)
\
zsr-9
Sx 2 -12x-9
o(;l
®
Remember you need to always list the restrictions before you reduce!!
And, you can never reduce (simply) through addition or subtraction.
2+5
2
-
therefore
--:;t:)
Example:
x+4
x-3
4
!!!
--:;t:-
-3
When multiplying and dividing rational expressions you will be using the same rules that apply to
multiplying and dividing rational numbers (fractions).
Remember that when you multiply rational numbers, you can simplify at the beginning or the end, and the
product is the same.
I
i
I
Rational Numbers
_g_
Method A
..4_ . 5x2 = 1 Ox3
120x2
15x2 8
_ 1x
-
1
-12
I
Method B
§_ = _1Q_
120
8
0
15
Rational Expressions Involving Variables
~
3
1
1
1
0
$=
-12
112
4
1
2x . firX'l
~ ,B
3
=~
12
4
***Remember when multiplying or dividing always list the restrictions***
When multiplying rational expressions:
1.
2.
3.
Factor all polynomials.
Give restrictions.
Simplify the factors. Make sure you are simplifying correctly!
84
H Secondary
Unit4
~
Example 4: Multiply the rational expressions and simplify all answers completely.
-l~
Ji_J
Restrictions:
C.
Restrictions:
X f- 0 1"3
Restrictions:
f -=/ -C? 1 0 1t \
x+S
x- 3
~ 2 4X+3 ~
('f.. -3i.Y-- 9 l\~ l"SJ
Restrictions:
'f1
~ 5\ \ 13
85
H Secondary 3
Unit4
When dividing rational expressions:
1. Factor all polynomials and give restrictions.
2. Multiply by the reciprocal (to divide the fractions) and give any n~ restrictions.
3. Simplify factors.
-
Example 5: Divide the rational expressions and simplify all answers completely.
a. ~bz + ~ •
4c
SrA.Q
.J&f 'L"'J.-?1~c
'l-
Restrictions:
0\.:f 0 1'r; i 0 1 (, 4 0
Restrictions:
'f.1 t 3 10
-..._
_,_
x 2 -4
_,_ x - 2
x2y2-xy · 3x2+19x-14 · xy
4x
d.
l\ 'f.
l ~~l)\3~2) . ~
:t1~~'j-\)• (Hl-}:~--2.)
(}-"~
3~~"4)L~---4)
c;)'l~~~)
L\1l).tt)( ~X/~
'2.
hy-\')l'f..t~L~-bJ
Restrictions:
1 t '-\
Y,.
1
0 \ .C? J~
Restrictions:
86

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