section 6.1 rational expression & functions: definitions, multiplying, dividing
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6.1 1
Section 6.1 Rational Expression & Functions:Definitions, Multiplying, Dividing
Fractions - a Quick Review Definitions:
Rational Functions, Expressions Finding the Domains (and Exclusions) of
Rational Functions Simplifying Rational Functions Simplifying by factoring out -1
6.1 2
Fractions - Review Q: When can you add or subtract fractions?
A: Only when denominators are the same Q: What do you do when denominators are not the same?
A: Use their LCD to create equivalent fractions. Q: How do you multiply fractions?
A: Factor all tops and factor all bottoms, cancel matching factors, multiply tops and bottoms
Q: What do you do first when dividing fractions? A: Turn division into multiplication : reciprocal the divisor.
Rational Expressions are Polynomial Fractions ! Same rules!
6.1 3
Definitions
6.1 4
Finding the Domain (and exclusions) of a Rational Function
Recall the domain of a function is the set ofall real numbers for which the function is defined. - What real values make this function undefined
(divided by 0)?
Factor: x2 + 2x – 24 = (x – 4)(x + 6) {x | x is Real, except for 4 or -6}
6.1 5
Graphs of Rational Functions
2t + 5 ≠ 0
2t ≠ -5
t ≠ -5/2
t=-5/2 is an Asymptote
6.1 6
Definitions Horizontal Asymptote – A horizontal line that
the graph of a function approaches as x values get very large or very small.
Vertical Asymptote – A vertical line that the graph of a function approaches as x values approach a fixed number
6.1 7
More Properties of Fractions - Review
6.1 8
2
3
22
23
4
24 4)3(
)4)(3(3
12ba
babbaa
abba
Simplifying Rational Expressions(In general, the expressions are NOT equivalent)
3)3(
)3)(3(392
x
xxx
xx
6.1 9
5)5()3(
)5)(5()3)(5(
25152
2
2
xx
xxx
xxx
xx
First Factor and Identify domain exclusions,Then Simplify
33
1)93)(3(
)93(27
932
2
3
2
x
xxxxxx
xxx
3,22
2)2)(2)(3(
)2)(3(21243
122223
2
xxxxx
xxxxx
xx
6.1 10
Multiplying Fractions
(First find domain exclusions)Factor expressions,
then cancel like factors
3,0)3()3(
)3(3
96 23232
aaaaa
aaaa
aaa
6.1 11
Example – Step by Step
4)3(
)3)(20()5)(3)(3(
)3)(20()5)(96(
35
2096 22222
xx
xxxxx
xxxxx
xx
xxx
1
4 1 1
1 x
1. Write down original problem2. Combine with parentheses3. Find any polynomials that need factoring4. Rewrite (if any factoring was done)5. Identify domain exclusions6. Cancel out matching factors7. Simplify the answer
x≠0,3
6.1 12
Board Practice – Rational Multiplication
1. Write original problem2. Combine w/ parens3. Factor polynomials4. Rewrite (if any factoring)5. Identify domain exclusions6. Cancel matching factors7. Simplify the answer
6456
4956
2
2
2
2
aaa
aaa
bxxbxxxx
22)2(
22
5
483 3m
bamba
6.1 13
Finding Powers of Rational Expressions Factor and Simplify (if possible) before applying the power If part of a larger expression, see if any terms cancel out Multiply out the terms in the numerator,
multiply out the terms in the denominator. Leave in simplified factored form
22
222
2 )6()5(
)6()6()5)(5(
)6(5
65
xxx
xxxxxx
xxx
xxx
6.1 14
Dividing FractionsChange Divide to
Multiply by Reciprocal,follow multiply procedure
23
4213
1422
4233
18
3342
18
2
2
2
23
2
23
xxxx
xxx
xxxxx
xxx
xxx
xxx
x
6.1 15
Board Practice - Rational Division
2242
448
2
23
xxx
xx
)2(143
b
xbb
5
483 3m
bamba
1. Write original problem2. Combine w/ parens3. Factor polynomials4. Rewrite (if any factoring)5. Identify domain exclusions6. Cancel matching factors7. Simplify the answer
6.1 16
Mixed Operations Multiplications & Division are done left to right In effect, make each divisor into a reciprocal
22
)2)(3()2)(3(
)2(1
)3(1)6()2()3()6( 22
xx
xxxx
xxxxxxxx
6.1 17
What Next? Present Section 6.2 Add/Subtract Rational Expressions
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