§6.1 Assignment
Notes
1. Exercises from 6.1
2. Read §6.2, 6.3.
M 1010 §6.1
(Math 1010)
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§6.1 - Rational Expressions and Functions
Notes
The ratio of polynomials are rational expressions. Some examples below:
1
2x
x 2 − 5x + 6
2x 3 − 6x + 1
,
,
, and
x3
x2 − 4
x 2 + 2x − 3
6x 2
Rational functions described as functions are rational functions with an
implied domain.
4
x −2
has the domain of all real numbers x such that x 6= 2. In interval notation,
you can write this as
f (x) =
Domain = (−∞, 2) ∪ (2, ∞)
(Math 1010)
M 1010 §6.1
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§6.1 - Finding the Domain
Notes
5t
# 15: Graph f (t) = 2
and find the domain.
t − 16
Make a table of t and f (t) values to graph f .
The denominator is zero when t 2 − 16 = 0. The rational expression is
undefined at those points. Solving this equation finds t = 4 or t = −4.
The domain is all real values x such that x 6= 4 and x 6= −4.
(Math 1010)
M 1010 §6.1
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§6.1 - Simplifying Rational Expressions
Notes
Simplified or reduced form is a rational expression whose numerator and
denominator have no common factors. Factor each, then divide out by
common factors.
1. Completely factor the numerator and denominator.
2. Divide out common factors to both the numerator and denominator.
Tip: Divide out by factors, not terms.
Examples:
2·2
2
(1)
=
2(x + 5)
(x + 5)
(2)
3+x
no common factors divide out
3 + 2x
M 1010 §6.1
(Math 1010)
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§6.1 Examples - Finding Domains
Notes
4
# 3: Find the domain of f (x) =
x −3
# 14: Find the domain of g (x) =
x3
x(x + 2)
Application: Example 3
You have started a small business that manufactures
lamps. The initial investment is $120,000. The cost of
manufacturing each lamp is $15. What is the average
cost C̄ (x) of producing x lamps, and what is the domain
of C̄ (x)?
M 1010 §6.1
(Math 1010)
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§6.1 Examples - Simplifying Rational Expressions
# 51: Simplify
x 2 (x − 8)
x(x − 8)
# 61: Simplify
y 3 − 4y
5xy + 3x 2 y 2
# 75: Simplify
y 2 + 4y − 12
xy 3
(Math 1010)
M 1010 §6.1
Notes
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