Absolute ValuesReview

Defn: A non-directional distance from zero.

Absolute values are always non-negative.

Does not exist.

4x x 4 4, and

21x x 21 21, and

7x

Absolute ValuesAbsolute Values of Expression

What value(s) of x will make the expression equal to 8 or -8?

843 x

123 x 43 x

843 x843 x

3

4x4x

4,3

4: xsolution

Absolute ValuesAbsolute Values of Expressions

What value(s) of x will make the expression equal to 7 or -7?

735 x

45 x 105 x

735 x735 x

2x5

4x

2,5

4: xsolution

Absolute ValuesAbsolute Values of Expressions

What value(s) of x will make the expression equal to 0?

032 x

32 x

032 x

2

3x

Absolute ValuesAbsolute Values of Expressions

What value(s) of x will make the expression equal to -1?

166 x

56 x

166 x

6

5x

166

56

165

11

11

No solution

166 x

166 x

6

7x

76 x

166

76

167

11

11

No solution

Absolute ValuesAbsolute Values of Inequalities

What value(s) of x will make the inequality greater than or equal to 8?

843 x

123 x

843 x

4x

843 x

843 x

43 x

3

4xInterval Notation

,4

3

4,

Absolute ValuesAbsolute Values of Inequalities

What value(s) of x will make the inequality greater than or equal to 8?

843 x

123 x

843 x

4x

843 x

843 x

43 x

3

4xInterval Notation

4,

3

4

Absolute ValuesAbsolute Values of Other Inequalities

2432 x

33 x

632 x

33 x

6x

Interval Notation

,60,

33 x

33 x

0x

Absolute ValuesAbsolute Values of Other Inequalities

532 xx

82 xx

532 xx

8x23 x

3

2x

532 xx

532 xx

22 xx

Graphs, Relations, and FunctionsDefn: A relation is a set of ordered pairs.

2,4,2,4,1,1,1,1,0,0 A

:Arange

4,1,0

2,1,0,1,2

Domain: The values of the 1st component of the ordered pair.

:Adomain

Range: The values of the 2nd component of the ordered pair.

The Rectangular Coordinate System Ordered Pair

(x, y)

(independent variable, dependent variable)

(1st component, 2nd component)

(input, output)

(abscissa, ordinate)

x

y

1st Quadrant2nd Quadrant

3rd Quadrant 4th Quadrant

The Rectangular Coordinate System

,x y ,x y

,x y ,x y

x

y

The Rectangular Coordinate System 2,4,2,4,1,1,1,1,0,0 A

Graphs, Relations, and FunctionsDefn: A function is a relation where every x value has one and only one value of y assigned to it.

:range 6,5,4,3

3,2,1,4:domain

x y

1 3

2 5

-4 6

1 4

3 3

x y

4 2

-3 8

6 1

-1 9

5 6

x y

2 3

5 7

3 8

-2 -5

8 7

:range 9,8,6,2,1

6,5,4,1,3 :domain

:range 8,7,3,5

8,5,3,2,2:domain

Graphs, Relations, and FunctionsFunctions and Equations.

32 xy

x y

0 -3

5 7

-2 -7

4 5

3 3

x y

2 4

-2 4

-4 16

3 9

-3 9

x y

1 1

1 -1

4 2

4 -2

0 0

2xy 2yx

function function not a function

x

y

The Rectangular Coordinate System 32 xy

x y

0 -3

5 7

-2 -7

4 5

3 3

x

y

The Rectangular Coordinate System 2xy

x y

2 4

-2 4

-4 16

3 9

-3 9

x

y

The Rectangular Coordinate System 2yx

x y

1 1

1 -1

4 2

4 -2

0 0

x

y

The Rectangular Coordinate System

Graphs, Relations, and FunctionsFunction Notation

52 xxf

13 xxf 13 xxy

xfxyy

Shorthand for stating that an equation is a function.

13 xy

Function notation also defines the value of x that is to be use to calculate the corresponding value of y.

Defines the independent variable (usually x) and the dependent variable (usually y).

5323 f 13 f 1,3

Graphs, Relations, and FunctionsFunction Notation – Restricted Domains

24 xxfy

42 xxfy

0x

042 x

x

xfy4

04 2 x