1.3 and 1.4 january 6, 2010. 1.1 p. 10-12 #2-12 even, 20-36 even 2) rational, real 4) natural,...
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1.3 and 1.4
January 6, 2010
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1.1 p. 10-12 #2-12 even, 20-36 even
2) rational, real4) natural, integer, rational, real6) Rational, real8) integer, rational, real
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1.1 continued
10) 3/1 (natural, integer, rational, real) -5/8 (rational, real)
√7 (irrational, real) .45… (rational, real)
0 (integer, rational, real) 5.6 x 103 (rational, real, integer, natural)
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1.1 continued
12) -103 (integer, rational, real)21/25 (rational, real)√100 = 10 (natural, integer, rational, real)-5.7/10 (rational, real)2/9 (rational, real)-1.457 (rational, real)√3 (irrational, real)
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1.1 20-36 even, 52-78 even
20) 1396.7%22) -4.9%24) 1.26237 x 105 26) 1.369 x 10-3 28) 1.056 x 102
30) -4.56 x 10-3 32) 1.25 x 106
34) 7 x 10-5
36) 7.89 x 10-1
52) 9 x 105, 900,000 54) 5.6 x 10-1, 0.5656) 6.25 x 10-4, 0.00062558) 3 x 102, 30060) 4.1 x 104, 41,00062) 5 x 101, 5064) 3.36 x 1019
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52-78 even, 84-90 even
66) 5 x 10-2
68) 3.78 x 102
70) 7.93872) -0.11674) 1.47276) 18.19678) 3.61884) 102,000
86) The Amazon river discharges more than 1 cubic mile per day
88) 0.0002 cm90) a) military decreased
between 1975 and 2002, with a slight increase between 1980-1985, and it held steady between 2000-2002
b) 2.6 million in 1975, 1.4 million in 2001
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1.2 (p. 25) #5-12 all, 21-28 all, 43-48 all, 61-68 all
5) a) max: 61, min: -30b) mean: 136/9= 15.1Median: 15, range: 91
6) a) max: 4.75, min: -3.5b) mean: 10.25/7=1.46Median: 1.5, range: 8.25
7) a) max: π2, min: √15 b) mean: 5.95Median: 4.51, range: 6.00
8) a) max: 31.2, min: 3√28 b) mean: 3.27Median: 3.14, range: 0.70
9) a) S={ (-1,5), (2, 2), (3, -1), (5, -4), (9, -5)
b) D= {-1, 2, 3, 5, 9)R= {5, 2, -1, -4, -5)
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1.2 (p. 25) #5-12 all, 21-28 all, 43-48 all, 61-68 all
10) a) S= {(-2, -4), (0, -2), (2, -1), (4, 0), (6, 4)}
b) D= {-2, 0, 2, 4, 6)R= {-4, -2, -1, 0, 4}
11) a) S= {(1, 5), (4, 5), (5, 6), (4, 6), (1, 5)}
b) D= {1, 4, 5}R= {5, 6}
12) a) S= {(-1, ½), (0, 1), (3, ¾), (-1, 3), (-2, -5/6)
b) D= {-2, -1, 0, 3}
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1.2 (p. 25) #5-12 all, 21-28 all, 43-48 all, 61-68 all
21) 5 22) 13
23) √29 = 5.39
24) √50= 7.07
25) √133.37= 11.55
26) √41.49= 6.44
27) 8
28) 8
43) 78.45 years
44) 1,082,905 inmates
45) 543,949 minutes
46) 207 million
47) 10 second
48) (a+b)/2
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1.2 (p. 25) #5-12 all, 21-28 all, 43-48 all, 61-68 all
61) D= {-3, -2, 0, 7)R= {-5, -3, 0, 4, 5}
62) D={-5, 0, 1, 3, 8}R= {-5, -2, 0, 1, 3}
63) D= {-4, -3, -1, 0, 2)R= {-2, -1, 1, 2, 3}
64) D= {-1, 1, 2}R= {-3, -1, 0, 1, 2}
65) D= {-35, -25, 0, 10, 75}R= {-55, -25, 25, 45, 50}
66) D= {-17, -7, -5, 2, 11}R= {-14, -13, -4, 15, 19}
67) D= {-0.7,0.1, 0.5, 0.8}R= {-0.3, -0.1, 0, 0.4}
68) D= {-1.2, -0.8, -0.3, 1.0}R= {-1.3, 0.5, 1.1, 1.5}
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Objectives
• Learn function notation• Represent a function four different ways• Identify the range and domain of a function• Identify functions
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What is a function?
• A function is a relation in which every input (x) has exactly one output (y).
• To emphasize that y is a function of x, the notation y=f(x) is often used.
• The variable y is called the dependent variable.
• The variable x is called the independent variable.
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Anatomy of a function
y= f(x)
• f(20)=4 would read..…f of 20 equals 4
• f, g, and h are often used as names of functions.
output
name
input
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Domain and range
• The set of valid or meaningful inputs (x) is called the domain of a function.
• The corresponding outputs (y) is the range.• Examine f(x)=x2.– What would be the domain?– What would be the range
• Unless stated otherwise, the domain of a function f is the set of all real numbers that make sense for the function. This is called the implied domain.
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Formal definition of a function
• A function is a relation in which each element in the domain corresponds to one element in the range.
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The vertical line test
• If every possible vertical line intersects a graph at no more than one point, then the graph represents a function.
• Do these pass?
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Representations of functions
• Verbally: in words• Numerically: table of values– Since it is often not possible or appropriate to list
all values, this type of table is often referred to as a partial numerical representation
• Symbolically: formula• Graphically: graph
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Using these representations
• Symbolic: f(x)=x+1Numerically Graphically
Verbally
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Find the domain of function f
f(x) = x2-4x-2
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Try this
• Find the domain and range.
• Evaluate f(0) and f(2)• Find all x such that
f(x)= -1
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1.4: Types of Functions and their Rates of Change
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Objectives
• Identify and use constant and linear functions.• Interpret slope as a rate of change.• Identify and use non-linear functions.• Recognize linear and non-linear data.• Use and interpret average rate of change.
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Constant FunctionA constant function is represented by:
f(x)=bwhere b is a constant (fixed) number.Examples include:
f(x)=10f(x)= ½
What would the graph of a constant function look like?
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Linear Function
A linear function is represented by:
f(x)=ax+bwhere a and b are constant (fixed) numbers.You may have also have seen this as y=mx+b.What would the graph of a linear function look like?
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Slope
• Slope = m = y2-y1
x2-x1
• Slope is also called rate of change.
• What is the slope of this graph?
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Linear vs. nonlinear
• These are examples of nonlinear graphs.
• Notice that none of them are straight lines.
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Average rate of change
• To find the average rate of change between two point, use the formula for slope.
• Plug in the two coordinates as x1, x2, y1, and y2.
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Find the average rate of
• Find the average rate of change between 0 and 4 for each graph.
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Your assignment
1.3 Page 42-45 #20-50#53-58#63-72
1.4, p. 58 #1-16, 25-30, 65-67, 75-78