1.3 and 1.4

January 6, 2010

1.1 p. 10-12 #2-12 even, 20-36 even

2) rational, real4) natural, integer, rational, real6) Rational, real8) integer, rational, real

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)

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)

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

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

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)

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}

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

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}

Objectives

• Learn function notation• Represent a function four different ways• Identify the range and domain of a function• Identify functions

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.

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

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.

Formal definition of a function

• A function is a relation in which each element in the domain corresponds to one element in the range.

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?

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

Using these representations

• Symbolic: f(x)=x+1Numerically Graphically

Verbally

Find the domain of function f

f(x) = x2-4x-2

Try this

• Find the domain and range.

• Evaluate f(0) and f(2)• Find all x such that

f(x)= -1

1.4: Types of Functions and their Rates of Change

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.

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?

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?

Slope

• Slope = m = y2-y1

x2-x1

• Slope is also called rate of change.

• What is the slope of this graph?

Linear vs. nonlinear

• These are examples of nonlinear graphs.

• Notice that none of them are straight lines.

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.

Find the average rate of

• Find the average rate of change between 0 and 4 for each graph.

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