Chapter 6 Rational Expressions, Functions, and Equations

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Chapter 6Rational Expressions, Functions, and
Equations
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6.1
Rational Expressions and Functions Multiplying
and Dividing
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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
Blitzer, Intermediate Algebra, 5e Slide 3
Section 6.1
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Rational Expressions
EXAMPLE
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 functions equation to solve the following
problem. Find and interpret f (60). Identify
your solution as a point on the graph.
p 393
Blitzer, Intermediate Algebra, 5e Slide 4
Section 6.1
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Rational Expressions
CONTINUED
Blitzer, Intermediate Algebra, 5e Slide 5
Section 6.1
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Rational Expressions
CONTINUED
SOLUTION
We use substitution to evaluate a rational
function, just as we did to evaluate other
functions in Chapter 2.
This is the given rational function.
Replace each occurrence of x with 60.
Perform the indicated operations.
Blitzer, Intermediate Algebra, 5e Slide 6
Section 6.1
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Rational Expressions
CONTINUED
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.
(60,195)
Blitzer, Intermediate Algebra, 5e Slide 7
Section 6.1
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Rational Expressions - Domain
EXAMPLE
Find the domain of f if
SOLUTION
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.
Set the denominator equal to 0.
Factor.
p 393
Blitzer, Intermediate Algebra, 5e Slide 8
Section 6.1
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Rational Expressions - Domain
CONTINUED
Set each factor equal to 0.
or
Solve the resulting equations.
Because 4 and 9 make the denominator zero, these
are the values to exclude. Thus,
or
Blitzer, Intermediate Algebra, 5e Slide 9
Section 6.1
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Rational Expressions - Domain
CONTINUED
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.
p 395
Blitzer, Intermediate Algebra, 5e Slide 10
Section 6.1
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Rational Expressions
Asymptotes Asymptotes
Vertical Asymptotes A 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 Expressions
1) 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 11
Section 6.1
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Rational Expressions - Domain
Check Point 2
Find the domain of f if
SOLUTION
Set the denominator equal to 0.
Factor.
Set each factor equal to 0.
or
Solve the resulting equations.
or
p 394
Blitzer, Intermediate Algebra, 5e Slide 12
Section 6.1
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Simplifying Rational Expressions
EXAMPLE
Simplify
SOLUTION
Factor the numerator and denominator.
Divide out the common factor, x 1.
Simplify.
Blitzer, Intermediate Algebra, 5e Slide 13
Section 6.1
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Simplifying Rational Expressions
Check Point 3
Simplify
SOLUTION
Factor the numerator and denominator.
Divide out the common factor, x 1.
Simplify.
p 397
Blitzer, Intermediate Algebra, 5e Slide 14
Section 6.1
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Simplifying Rational Expressions
EXAMPLE
Simplify
SOLUTION
Factor the numerator and denominator.
Rewrite 3 x as (-1)(-3 x).
Rewrite -3 x as x 3.
Blitzer, Intermediate Algebra, 5e Slide 15
Section 6.1
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Simplifying Rational Expressions
CONTINUED
Divide out the common factor, x 3.
Simplify.
Do Check 4a and 4b on page 397
Blitzer, Intermediate Algebra, 5e Slide 16
Section 6.1
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Multiplying Rational Expressions
Multiplying Rational Expressions
1) 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 17
Section 6.1
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Multiplying Rational Expressions
EXAMPLE
Multiply
SOLUTION
This is the original expression.
Factor the numerators and denominators completely.
Divide numerators and denominators by common
factors.
Blitzer, Intermediate Algebra, 5e Slide 18
Section 6.1
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Multiplying Rational Expressions
CONTINUED
Multiply the remaining factors in the numerators
and in the denominators.
Note that when simplifying rational expressions
or multiplying rational expressions, we just used
factoring. With one additional step that is
provided in the following Definition for
Division, division of rational expressions
promises to be just as straightforward.
Blitzer, Intermediate Algebra, 5e Slide 19
Section 6.1
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Multiplying Rational Expressions
EXAMPLE
Multiply
SOLUTION
This is the original expression.
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.
Blitzer, Intermediate Algebra, 5e Slide 20
Section 6.1
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Multiplying Rational Expressions
CONTINUED
Now you may multiply the remaining factors in the
numerators and in the denominators.
or
Blitzer, Intermediate Algebra, 5e Slide 21
Section 6.1
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Multiplying Rational Expressions
Check Point 5
Multiply
pages 398-399
Blitzer, Intermediate Algebra, 5e Slide 22
Section 6.1
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Multiplying Rational Expressions
Check Point 6
Multiply
pages 398-399
Blitzer, Intermediate Algebra, 5e Slide 23
Section 6.1
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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 Expressions
If P, Q, R, and S are polynomials, where then
Replace with its reciprocal by interchanging
its numerator and denominator.
Change division to multiplication.
Blitzer, Intermediate Algebra, 5e Slide 24
Section 6.1
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Dividing Rational Expressions
EXAMPLE
Divide
SOLUTION
This is the original expression.
Invert the divisor and multiply.
Factor.
Blitzer, Intermediate Algebra, 5e Slide 25
Section 6.1
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Dividing Rational Expressions
CONTINUED
Divide numerators and denominators by common
factors.
Multiply the remaining factors in the numerators
and in the denominators.
Blitzer, Intermediate Algebra, 5e Slide 26
Section 6.1
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Multiplying Rational Expressions
Check Point 7a
Divide
Blitzer, Intermediate Algebra, 5e Slide 27
Section 6.1
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Multiplying Rational Expressions
Check Point 7b
Divide
Blitzer, Intermediate Algebra, 5e Slide 28
Section 6.1
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DONE