Post on 17-Dec-2015
transcript
Chapter 6Rational Expressions,
Functions, and Equations
§ 6.1
Rational Expressions and Functions: Multiplying and Dividing
Blitzer, Intermediate Algebra, 5e – Slide #3 Section 6.1
Rational Expressions
A rational expression consists of a polynomial divided by a nonzero polynomial (denominator cannot be equal to 0).
A rational function is a function defined by a formula that is a rational expression. For example, the following is a rational function:
x
xxf
100
130
Blitzer, Intermediate Algebra, 5e – Slide #4 Section 6.1
Rational Expressions
EXAMPLEEXAMPLE
The rational function
models the cost, f (x) in millions of dollars, to inoculate x% of the population against a particular strain of flu. The graph of the rational function is shown. Use the function’s equation to solve the following problem.
Find and interpret f (60). Identify your solution as a point on the graph.
x
xxf
100
130
p 393
Blitzer, Intermediate Algebra, 5e – Slide #5 Section 6.1
Rational Expressions
CONTINUECONTINUEDD
0
100
200
300
400
500
600
700
800
900
1000
0 20 40 60 80 100
Blitzer, Intermediate Algebra, 5e – Slide #6 Section 6.1
Rational Expressions
SOLUTIONSOLUTION
We use substitution to evaluate a rational function, just as we did to evaluate other functions in Chapter 2.
CONTINUECONTINUEDD
x
xxf
100
130 This is the given rational function.
60100
6013060
f
Replace each occurrence of x with 60.
19540
7800 Perform the indicated
operations.
Blitzer, Intermediate Algebra, 5e – Slide #7 Section 6.1
Rational Expressions
Thus, f (60) = 195. This means that the cost to inoculate 60% of the population against a particular strain of the flu is $195 million. The figure below illustrates the solution by the point (60,195) on the graph of the rational function.
CONTINUECONTINUEDD
0
100
200
300
400
500
600
700
800
900
1000
0 20 40 60 80 100
(60,195)
Blitzer, Intermediate Algebra, 5e – Slide #8 Section 6.1
Rational Expressions - Domain
EXAMPLEEXAMPLE
Find the domain of f if
.3613
32
xx
xxf
The domain of f is the set of all real numbers except those for which the denominator is zero. We can identify such numbers by setting the denominator equal to zero and solving for x.
SOLUTIONSOLUTION
036132 xx Set the denominator equal to 0.
094 xx Factor.p 393
Blitzer, Intermediate Algebra, 5e – Slide #9 Section 6.1
Rational Expressions - Domain
Because 4 and 9 make the denominator zero, these are the values to exclude. Thus,
9 and 4 andnumber real a is | ofDomain xxxxf
Set each factor equal to 0.04 x
Solve the resulting equations.
CONTINUECONTINUEDD
09 xor4x 9x
. ,99,4,4- ofDomain for
Blitzer, Intermediate Algebra, 5e – Slide #10 Section 6.1
Rational Expressions - Domain
CONTINUECONTINUEDD . ,99,4,4- ofDomain f
In this example, we excluded 4 and 9 from the domain. Unlike the graph of a polynomial which is continuous, this graph has two breaks in it – one at each of the excluded values. Since x cannot be 4 or 9, there is not a function value corresponding to either of those x values. At 4 and at 9, there will be dashed vertical lines called vertical asymptotes. The graph of the function will approach these vertical lines on each side as the x values draw closer and closer to each of them, but will not touch (cross) the vertical lines. The lines x = 4 and x = 9 each represent vertical asymptotes for this particular function.
.)9)(4(
3
xx
xxf
p 395
Blitzer, Intermediate Algebra, 5e – Slide #11 Section 6.1
Rational Expressions
AsymptotesVertical
AsymptotesA vertical line that the graph of a function approaches, but does not touch.
Horizontal Asymptotes
A horizontal line that the graph of a function approaches as x gets very large or very small. The graph of a function may touch/cross its horizontal asymptote.
Simplifying Rational Expressions1) Factor the numerator and the denominator completely.
2) Divide both the numerator and the denominator by any common factors.
p 395
Blitzer, Intermediate Algebra, 5e – Slide #12 Section 6.1
Rational Expressions - Domain
Check Point 2Check Point 2
Find the domain of f if .352
52
xx
xxf
SOLUTIONSOLUTION
0352 2 xx Set the denominator equal to 0.
0312 xx Factor.
p 394
012 x
Solve the resulting equations.
03 x
2
1x 3x
Set each factor equal to 0.or
2
1 and 3 andnumber real a is | ofDomain xxxxf
. ,2
1
2
1,3,-3- ofDomain
for
Blitzer, Intermediate Algebra, 5e – Slide #13 Section 6.1
Simplifying Rational Expressions
EXAMPLEEXAMPLE
Simplify: .32
12
xx
x
SOLUTIONSOLUTION
Factor the numerator and denominator.
13
11
32
12
xx
x
xx
x
Divide out the common factor, x + 1.
13
11
xx
x
3
1
xSimplify.
Blitzer, Intermediate Algebra, 5e – Slide #14 Section 6.1
Simplifying Rational Expressions
Check Point 3Check Point 3
Simplify: .2
1072
x
xx
SOLUTIONSOLUTION
Factor the numerator and denominator.
21
25
2
1072
x
xx
x
xx
Divide out the common factor, x + 1.
21
25
x
xx
5x Simplify.
p 397
Blitzer, Intermediate Algebra, 5e – Slide #15 Section 6.1
Simplifying Rational Expressions
EXAMPLEEXAMPLE
Simplify: .9
1272
2
x
xx
SOLUTIONSOLUTION
Factor the numerator and denominator.
Rewrite 3 – x as (-1)(-3 + x).
xx
xx
x
xx
33
43
9
1272
2
xx
xx
313
43
313
43
xx
xxRewrite -3 + x as x – 3.
Blitzer, Intermediate Algebra, 5e – Slide #16 Section 6.1
Simplifying Rational Expressions
Divide out the common factor, x – 3.
Simplify.
313
43
xx
xx
CONTINUECONTINUEDD
13
4
x
x
Do Check 4a and 4b on page 397
Blitzer, Intermediate Algebra, 5e – Slide #17 Section 6.1
Multiplying Rational Expressions
Multiplying Rational Expressions1) Factor all numerators and denominators completely.
2) Divide numerators and denominators by common factors.
3) Multiply the remaining factors in the numerators and multiply the remaining factors in the denominators.
p 398
Blitzer, Intermediate Algebra, 5e – Slide #18 Section 6.1
Multiplying Rational Expressions
EXAMPLEEXAMPLE
Multiply: .6
23
43
23222
22
22
22
yxyx
yxyx
yxyx
yxyx
SOLUTIONSOLUTION
Factor the numerators and denominators completely.
Divide numerators and denominators by common factors.
yxyx
yxyx
yxyx
yxyx
32
3
3
22
22
22
22
22
6
23
43
232
yxyx
yxyx
yxyx
yxyx
This is the original expression.
yxyx
yxyx
yxyx
yxyx
32
3
3
22
Blitzer, Intermediate Algebra, 5e – Slide #19 Section 6.1
Multiplying Rational Expressions
Multiply the remaining factors in the numerators and in the denominators.
yxyx
yxyx
33
32
CONTINUECONTINUEDD
Note that when simplifying rational expressions or multiplying rational expressions, we just used factoring.
With one additional step that is provided in the followingDefinition for Division, division of rational expressions promises to be just as straightforward.
Blitzer, Intermediate Algebra, 5e – Slide #20 Section 6.1
Multiplying Rational Expressions
EXAMPLEEXAMPLE
Multiply: .4
3
9
2822 yy
y
y
y
SOLUTIONSOLUTION
Factor the numerators and denominators completely.
Divide numerators and denominators by common factors. Because 3 – y and y -3 are opposites, their quotient is -1.
This is the original expression.yy
y
y
y
22 4
3
9
28
14
3
33
142
yy
y
yy
y
14
)3(1
33
142
yy
y
yy
y
Blitzer, Intermediate Algebra, 5e – Slide #21 Section 6.1
Multiplying Rational Expressions
Now you may multiply the remaining factors in the numerators and in the denominators.
yy 3
2
CONTINUECONTINUEDD
or yy 3
2
Blitzer, Intermediate Algebra, 5e – Slide #22 Section 6.1
Multiplying Rational Expressions
Check Point 5Check Point 5
Multiply: .16
214
7
42
2
x
xx
x
x
4
3
x
x
pages 398-399
Blitzer, Intermediate Algebra, 5e – Slide #23 Section 6.1
Multiplying Rational Expressions
Check Point 6Check Point 6
Multiply: .49
443
36
842
2
2
x
xx
xx
x
233
24
xx
x 233
24
xx
x
pages 398-399
Blitzer, Intermediate Algebra, 5e – Slide #24 Section 6.1
Dividing Rational Expressions
Simplifying Rational Expressions with Opposite Factors in the Numerator and Denominator
The quotient of two polynomials that have opposite signs and are additive inverses is -1.
Dividing Rational ExpressionsIf P, Q, R, and S are polynomials, where then,0 and ,0 ,0 SRQ
.QR
PS
R
S
Q
P
S
R
Q
P
Change division to multiplication.
Replace with its reciprocal by interchanging its numerator and denominator.
S
R
Blitzer, Intermediate Algebra, 5e – Slide #25 Section 6.1
Dividing Rational Expressions
EXAMPLEEXAMPLE
Divide: .352
32
12 22
22
2
2
yxyx
yxyx
xx
yxy
SOLUTIONSOLUTION
Invert the divisor and multiply.
Factor.
This is the original expression.22
22
2
2
352
32
12 yxyx
yxyx
xx
yxy
22
22
2
2
32
352
12 yxyx
yxyx
xx
yxy
yxyx
yxyx
x
yxy
32
32
1 2
Blitzer, Intermediate Algebra, 5e – Slide #26 Section 6.1
Dividing Rational Expressions
Divide numerators and denominators by common factors.
yxyx
yxyx
x
yxy
32
32
1 2
CONTINUECONTINUEDD
21
x
yxy Multiply the remaining factors in the numerators and in the denominators.
Blitzer, Intermediate Algebra, 5e – Slide #27 Section 6.1
Multiplying Rational Expressions
Check Point 7aCheck Point 7a
Divide: .9
73499 2
x
x
739 x
Blitzer, Intermediate Algebra, 5e – Slide #28 Section 6.1
Multiplying Rational Expressions
Check Point 7bCheck Point 7b
Divide: .6
2410
5
122
22
xx
xx
x
xx
5
3
x
DONE