Transcript Sect. 2.2
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
1
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
2
Definitions
6.1
3
Finding the Domain (and exclusions) of a Rational Function Recall the
domain
of a function is the set of all real numbers for which the function is defined. -What real values make this function undefined (divided by 0)?
Factor: x 2 + 2x – 24 = (x – 4)(x + 6) {x | x is Real, except for 4 or -6} 6.1
4
Graphs of Rational Functions
t=-5/2 is an Asymptote 2t + 5 ≠ 0 2t ≠ -5 t ≠ -5/2 6.1
5
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
6
More Properties of Fractions - Review 6.1
7
Simplifying Rational Expressions
(In general, the expressions are NOT equivalent)
12
a
4
b
2 3
ab
4 ( 3 )( 4 )
aa
3
b
2 ( 3 )
ab
2
b
2 4
a b
2 3
x x
2 9 3 (
x
(
x
3 )(
x
3 ) 6.1
3 )
x
3 8
First Factor and Identify domain exclusions, Then Simplify
x
2 2
x
25
x
2 15 (
x
( 5 5 )(
x
3 )
x
)( 5
x
) (
x
3 ) ( 5
x
)
x
5
x
2
x
3 3
x
9 27 (
x
2 (
x
3 )(
x
2 3
x
9 ) 3
x
9 )
x
1 3
x
3
x
3 2
x
2 3
x
2 2
x
4 12
x
12
x
2 , 3 (
x
2 (
x
3 )(
x
3 )(
x
2 )(
x
2 ) 2 ) 6.1
x
2 2 9
Multiplying Fractions
a
2 6
a
9
a a a
3 3 (
a
3 ) 2
a
3
a
(
a
3 )
a
2 (
a
3 ) (First find domain exclusions) Factor expressions, then cancel like factors 6.1
a
0 , 3 10
Example – Step by Step
x≠0,3
x
2 6
x
9 20
x x
5
x
2 3 (
x
2 6
x
( 20
x
)( 9 )( 5
x
2 )
x
3 ) (
x
1 4 3 )( 1
x
( 20
x
)( 1 3 )( 5
x
x
2 )
x
1 3 )
x
(
x
3 ) 4 1.
2.
3.
4.
5.
6.
7.
Write down original problem Combine with parentheses Find any polynomials that need factoring Rewrite (if any factoring was done) Identify domain exclusions Cancel out matching factors Simplify the answer 6.1
11
Board Practice – Rational Multiplication
a
3
b m
8 3
a
4
b m
5 ( 2
x
x
2 )
x
2
xb x
2
x
2
b a
2
a a
2 56 49
a
2
a a
2 56 64 1.
2.
3.
Write original problem Combine w/ parens Factor polynomials 4.
5.
Rewrite (if any factoring) Identify domain exclusions 6.
Cancel matching factors 7.
6.1
Simplify the answer 12
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
x
2
x
5 6
x
2
x x
(
x
5 6 ) 2 (
x
5 )(
x
5 )
x
(
x
6 )
x
(
x
6 ) (
x
5 ) 2
x
2 (
x
6 ) 2 6.1
13
Dividing Fractions
Change Divide to Multiply by Reciprocal , follow multiply procedure
x
3
x x
1 8 2
x x
2
x
3 2
x
2 2 2
x
x
3
x
4 4
x
2 3
x x x
3
x
8 1
x
3 2 2
x
4
x
2 2 3
x x
3
x x
4 2 6.1
14
Board Practice - Rational Division
a
3
b
8
m
3
a
4
b m
5
x
4
x
3 8 4
x
2 2
x
2
x
2 2 4 1.
2.
3.
4.
5.
6.
7.
Write original problem Combine w/ parens Factor polynomials Rewrite (if any factoring) Identify domain exclusions Cancel matching factors Simplify the answer 6.1
b
3
x
4
b
1 (
b
2 ) 15
Mixed Operations
Multiplications & Division are done left to right In effect, make each divisor into a reciprocal (
x
2
x
6 ) (
x
3 ) (
x
2 ) (
x
2
x
6 ) (
x
1 3 ) (
x
1 2 ) (
x
(
x
3 )(
x
3 )(
x
2 ) 2 )
x x
2 2 6.1
16
What Next?
Present Section 6.2
Add/Subtract Rational Expressions 6.1
17