Introduction to Interval Notation

Goals:

I will be able to use inequality and interval notation to graph inequalities.

I will be able to use the graphs of inequalities to write inequality and interval notation.

I will determine the domain and range from a graph.

• Inequality Notation: uses inequality symbols to indicate the limits of a set of values.≤, ≥, Ex. -3

• Interval Notation is an alternative to expressing as an inequaility. Uses or [ , ]

• End values not included ( or ) or

• End values included [ or ] or

Use for or ) for

Inequality Interval GraphNotation Notation

-3 x 5

x 2

x -4 or x 2

Inequality Interval GraphNotation Notation

-3 x 5

x 2

x -4 or x 2

-3 -2 -1 0 1 2 3 4 5

-3 -2 -1 0 1 2 3 4 5

-4 -3 -2 -1 0 1 2 3 4 5

Inequality Interval GraphNotation Notation

-3 x 5 [ -3, 5 )

x 2

x -4 or x 2

-3 -2 -1 0 1 2 3 4 5

-3 -2 -1 0 1 2 3 4 5

-4 -3 -2 -1 0 1 2 3 4 5

Inequality Interval GraphNotation Notation

-3 x 5 [ -3, 5 )

x 2

x -4 or x 2

-3 -2 -1 0 1 2 3 4 5

-3 -2 -1 0 1 2 3 4 5

-4 -3 -2 -1 0 1 2 3 4 5

Inequality Interval GraphNotation Notation

-3 x 5 [ -3, 5 )

x 2 ( 2, )

x -4 or x 2

-3 -2 -1 0 1 2 3 4 5

-3 -2 -1 0 1 2 3 4 5

-4 -3 -2 -1 0 1 2 3 4 5

Inequality Interval GraphNotation Notation

-3 x 5 [ -3, 5 )

x 2 ( 2, )

x -4 or x 2

-3 -2 -1 0 1 2 3 4 5

-3 -2 -1 0 1 2 3 4 5

-4 -3 -2 -1 0 1 2 3 4 5

Inequality Interval GraphNotation Notation

-3 x 5 [ -3, 5 )

x 2 ( 2, )

x -4 or x 2 ( -, -4 ] (2, )

-3 -2 -1 0 1 2 3 4 5

-3 -2 -1 0 1 2 3 4 5

-4 -3 -2 -1 0 1 2 3 4 5

x

y

O 2

2

Domain and Range

Domain Domain

Range Range

x

y

O 2

2

x

y

O 2

2

Domain and Range

Domain [-2, 5) Domain

Range [0, 6] Range

x

y

O 2

2

x

y

O 2

2

Domain and Range

Domain [-2, 5) Domain (-, 4]

Range [0, 6] Range [-7, )

x

y

O 2

2

Attributes of Functions

Goals:

I will determine whether a function is positive or negative within a given interval.

I will determine whether a function is increasing or decreasing within a given interval.

I will determine whether a function is continuous or discrete.

The function is increasing where the y-values are getting larger from left to right (the graph rises from left to right).

The function is decreasing where the y-values are getting smaller from left to right (the graph falls from left to right).

Write increasing and decreasing intervals using the domain for that interval (x-values).

Increasing and decreasing intervals are always exclusive of the end points (i.e., always use parenthesis).

x

y

0

2

2

x

y

0

2

2

x

y

0

2

2

D

x

y

0

2

2

D I

x

y

0

2

2

Increasing:

x

y

0

2

2

Increasing:

x

y

0

2

2

Increasing:

(2.5, )

x

y

0

2

2

Increasing:

(2.5, )

Decreasing:

x

y

0

2

2

Increasing:

(2.5, )

Decreasing:

x

y

0

2

2

Increasing:

(2.5, )

Decreasing:

(- , 2.5)

x

y

0 2

2

x

y

0 2

2

Increasing:

Decreasing:

x

y

0 2

2

Increasing:

Decreasing:

x

y

0 2

2

Increasing:

Decreasing:

(2, 7)

x

y

0 2

2

Increasing:

Decreasing:

(2, 7)

x

y

0 2

2

Increasing:

Decreasing:

(2, 7)

(-3, 2)

x

y

O 2

2

x

y

O 2

2

Increasing:

Decreasing:

x

y

O 2

2

Increasing:

Decreasing:

x

y

O 2

2

Increasing:

Decreasing:

(-, -3)

x

y

O 2

2

Increasing:

Decreasing:

(-, -3)

x

y

O 2

2

Increasing:

Decreasing:

(-, -3) (1, )

x

y

O 2

2

Increasing:

Decreasing:

(-, -3) (1, )

x

y

O 2

2

Increasing:

Decreasing:

(-, -3) (1, )

(-3, 1)

Positive or negative intervals are notated using the domain for the interval.

Since zero is neither positive nor negative, if the interval ends where a y-value is zero, that point is excluded (parenthesis).

The function is positive where its y-values are positive.

The function is negative where its y-values are negative.

x

y

0 2

2

x

y

0 2

2

Positive:

Negative:

x

y

0 2

2

Positive:

Negative:

x

y

0 2

2

Positive:

Negative:

(-, )

x

y

0 2

2

Positive:

Negative:

(-, )

x

y

0 2

2

x

y

0 2

2

Positive:

Negative:

x

y

0 2

2

Positive:

Negative:

x

y

0 2

2

Positive:

Negative:

(-3, 0)

x

y

0 2

2

Positive:

Negative:

(-3, 0) U

x

y

0 2

2

Positive:

Negative:

(-3, 0) U (4, 7]

x

y

0 2

2

Positive:

Negative:

(-3, 0) U (4, 7]

x

y

0 2

2

Positive:

Negative:

(-3, 0) U (4, 7]

(0, 4)

x

y

O 2

2

x

y

O 2

2

Positive:

Negative:

x

y

O 2

2

Positive:

Negative:

x

y

O 2

2

Positive:

Negative:

(- 4.5, -1)

x

y

O 2

2

Positive:

Negative:

(- 4.5, -1) U

x

y

O 2

2

Positive:

Negative:

(- 4.5, -1) U (3, )

x

y

O 2

2

Positive:

Negative:

(- 4.5, -1) U (3, )

x

y

O 2

2

Positive:

Negative:

(- 4.5, -1) U (3, )

(-, -4.5)

x

y

O 2

2

Positive:

Negative:

(- 4.5, -1) U (3, )

(-, -4.5) U

x

y

O 2

2

Positive:

Negative:

(- 4.5, -1) U (3, )

(-, -4.5) U (-1, 3)

x

y

O 2

2

Identify the domain and range of the given relation. Then represent the relation using a graph and a mapping diagram.

{(-2, 3), (1, 2), (3, -1), (-4, -3)}

x

y

O 2

2

Identify the domain and range of the given relation. Then represent the relation using a graph and a mapping diagram.

{(-2, 3), (1, 2), (3, -1), (-4, -3)}

D: { -4, -2, 1, 3}

R: {-3, -1, 2, 3}

x

-4

-2

1

3

y

-3

-1

2

3

IS IT A FUNCTION? HOW DO YOU KNOW???

Sec 2.1- Extension

Use Discrete and Continuous Functions

Discrete function consists of separate, unconnected points in a plane

Continuous function consists of an unbroken line or curve or an unbroken portion of a line or curve.

Graph the function for the given domain. Classify the function as discrete or continuous for the domain, and then identify the range. (page 80)

1f(x) x 1

2

Domain: x = -2, 0, 2, 4

x

y

0 2

2

Graph the function for the given domain. Classify the function as discrete or continuous for the domain, and then identify the range. (page 80)

1f(x) x 1

2

Domain: x = -2, 0, 2, 4

Discrete points for domain, graph is

unconnected set of points

x

y

0 2

2

Graph the function for the given domain. Classify the function as discrete or continuous for the domain, and then identify the range. (page 80)

1f(x) x 1

2

Domain: x = -2, 0, 2, 4

Create a table with the given domain values

x

y

0 2

2

Graph the function for the given domain. Classify the function as discrete or continuous for the domain, and then identify the range. (page 80)

1f(x) x 1

2

Domain: x = -2, 0, 2, 4

x y-2024

x

y

0 2

2

Graph the function for the given domain. Classify the function as discrete or continuous for the domain, and then identify the range. (page 80)

1f(x) x 1

2

Domain: x = -2, 0, 2, 4

x y-2 00 12 24 3

x

y

0 2

2

Graph the function for the given domain. Classify the function as discrete or continuous for the domain, and then identify the range. (page 80)

1f(x) x 1

2

Domain: x = -2, 0, 2, 4

x y-2 00 12 24 3

Plot the points

x

y

0 2

2

Graph the function for the given domain. Classify the function as discrete or continuous for the domain, and then identify the range. (page 80)

1f(x) x 1

2

Domain: x = -2, 0, 2, 4

x

y

0 2

2

x y-2 00 12 24 3

Discrete graph

Graph the function for the given domain. Classify the function as discrete or continuous for the domain, and then identify the range. (page 80)

1f(x) x 1

2

Domain: x = -2, 0, 2, 4

x

y

0 2

2

x y-2 00 12 24 3

Graph the function for the given domain. Classify the function as discrete or continuous for the domain, and then identify the range. (page 80)

1f(x) x 1

2

Domain: x -3

x

y

0 2

2

x y-3

Graph the function for the given domain. Classify the function as discrete or continuous for the domain, and then identify the range. (page 80)

1f(x) x 1

2

Domain: x -3

x

y

0 2

2

Graph the function for the given domain. Classify the function as discrete or continuous for the domain, and then identify the range. (page 80)

1f(x) x 1

2

Domain: x -3

x

y

0 2

2

Continuous set of points for domain, graph is a connected set of points

Graph the function for the given domain. Classify the function as discrete or continuous for the domain, and then identify the range. (page 80)

1f(x) x 1

2

Domain: x -3

x

y

0 2

2

Create a table with the given domain value

Graph the function for the given domain. Classify the function as discrete or continuous for the domain, and then identify the range. (page 80)

1f(x) x 1

2

Domain: x -3

x

y

0 2

2

x y-3 -.5

Graph the function for the given domain. Classify the function as discrete or continuous for the domain, and then identify the range. (page 80)

1f(x) x 1

2

Domain: x -3

x

y

0 2

2

x y-3 -.5

Plot the point; graph is unbroken; it is a ray with

endpoint at (-3, -.5)

Graph the function for the given domain. Classify the function as discrete or continuous for the domain, and then identify the range. (page 80)

1f(x) x 1

2

Domain: x -3

x

y

0 2

2

x y-3 -.5

Graph the function for the given domain. Classify the function as discrete or continuous for the domain, and then identify the range. (page 80)

1f(x) x 1

2

Domain: x -3

x

y

0 2

2

x y-3 -.5

Continuous graph

x

y

0 2

2

Write and graph the function described. Determine the domain and range. Then tell whether the function is discrete or continuous.

A student group is selling chocolate bars for $2 each. The function f(x) gives the amount of money collected after selling x chocolate bars.

x

y

0 2

2

Write and graph the function described. Determine the domain and range. Then tell whether the function is discrete or continuous.

A student group is selling chocolate bars for $2 each. The function f(x) gives the amount of money collected after selling x chocolate bars.

f(x) = 2x

x

y

0 2

2

Write and graph the function described. Determine the domain and range. Then tell whether the function is discrete or continuous.

A student group is selling chocolate bars for $2 each. The function f(x) gives the amount of money collected after selling x chocolate bars.

f(x) = 2x

x y-31.5

Negative values are not reasonable;

Fractions are not reasonable

x

y

0 2

2

Write and graph the function described. Determine the domain and range. Then tell whether the function is discrete or continuous.

A student group is selling chocolate bars for $2 each. The function f(x) gives the amount of money collected after selling x chocolate bars.

f(x) = 2x

x y0123

x

y

0 2

2

Write and graph the function described. Determine the domain and range. Then tell whether the function is discrete or continuous.

A student group is selling chocolate bars for $2 each. The function f(x) gives the amount of money collected after selling x chocolate bars.

f(x) = 2x

x y0 01 22 43 6

x

y

0 2

2

Write and graph the function described. Determine the domain and range. Then tell whether the function is discrete or continuous.

A student group is selling chocolate bars for $2 each. The function f(x) gives the amount of money collected after selling x chocolate bars.

f(x) = 2x

x y0 01 22 43 6

Domain:Range:

x

y

0 2

2

Write and graph the function described. Determine the domain and range. Then tell whether the function is discrete or continuous.

A student group is selling chocolate bars for $2 each. The function f(x) gives the amount of money collected after selling x chocolate bars.

f(x) = 2x

x y0 01 22 43 6

Domain: Whole numbers 0; {0, 1, 2,…}Range:

x

y

0 2

2

Write and graph the function described. Determine the domain and range. Then tell whether the function is discrete or continuous.

A student group is selling chocolate bars for $2 each. The function f(x) gives the amount of money collected after selling x chocolate bars.

f(x) = 2x

x y0 01 22 43 6

Domain: Whole numbers 0; {0, 1, 2,…}Range: Positive multiples of 2; {0, 2, 4,…}

x

y

0 2

2

Write and graph the function described. Determine the domain and range. Then tell whether the function is discrete or continuous.

A student group is selling chocolate bars for $2 each. The function f(x) gives the amount of money collected after selling x chocolate bars.

f(x) = 2x

x y0 01 22 43 6

Domain: Whole numbers 0; {0, 1, 2,…}Range: Positive multiples of 2; {0, 2, 4,…}

x

y

0 2

2

Write and graph the function described. Determine the domain and range. Then tell whether the function is discrete or continuous.

A student group is selling chocolate bars for $2 each. The function f(x) gives the amount of money collected after selling x chocolate bars.

f(x) = 2x

x y0 01 22 43 6

Domain: Whole numbers 0; {0, 1, 2,…}Range: Positive multiples of 2; {0, 2, 4,…} Discrete graph

x

y

0 2

2

Write and graph the function described. Determine the domain and range. Then tell whether the function is discrete or continuous.

A low-flow shower head releases 1.8 gallons of water per minute. The function V(x) gives the volume of water released after x minutes.

x

y

0 2

2

Write and graph the function described. Determine the domain and range. Then tell whether the function is discrete or continuous.

A low-flow shower head releases 1.8 gallons of water per minute. The function V(x) gives the volume of water released after x minutes.

f(x) = 1.8x

x

y

0 2

2

Write and graph the function described. Determine the domain and range. Then tell whether the function is discrete or continuous.

A low-flow shower head releases 1.8 gallons of water per minute. The function V(x) gives the volume of water released after x minutes.

f(x) = 1.8x

Negative values are not reasonable;

Fractions are reasonable

x y-31.5

x

y

0 2

2

Write and graph the function described. Determine the domain and range. Then tell whether the function is discrete or continuous.

A low-flow shower head releases 1.8 gallons of water per minute. The function V(x) gives the volume of water released after x minutes.

f(x) = 1.8x

x y0.51.22.5

x

y

0 2

2

Write and graph the function described. Determine the domain and range. Then tell whether the function is discrete or continuous.

A low-flow shower head releases 1.8 gallons of water per minute. The function V(x) gives the volume of water released after x minutes.

f(x) = 1.8x

x y0 0.5 .92.5 4.55 9

x

y

0 2

2

Write and graph the function described. Determine the domain and range. Then tell whether the function is discrete or continuous.

A low-flow shower head releases 1.8 gallons of water per minute. The function V(x) gives the volume of water released after x minutes.

f(x) = 1.8x

x y0 0.5 .92.5 4.55 9

Domain:Range:

x

y

0 2

2

Write and graph the function described. Determine the domain and range. Then tell whether the function is discrete or continuous.

A low-flow shower head releases 1.8 gallons of water per minute. The function V(x) gives the volume of water released after x minutes.

f(x) = 1.8x

x y0 0.5 .92.5 4.55 9

Domain: x 0Range:

x

y

0 2

2

Write and graph the function described. Determine the domain and range. Then tell whether the function is discrete or continuous.

A low-flow shower head releases 1.8 gallons of water per minute. The function V(x) gives the volume of water released after x minutes.

f(x) = 1.8x

x y0 0.5 .92.5 4.55 9

Domain: x 0Range: y 0

x

y

0 2

2

Write and graph the function described. Determine the domain and range. Then tell whether the function is discrete or continuous.

A low-flow shower head releases 1.8 gallons of water per minute. The function V(x) gives the volume of water released after x minutes.

f(x) = 1.8x

x y0 0.5 .92.5 4.55 9

Domain: x 0Range: y 0

x

y

0 2

2

Write and graph the function described. Determine the domain and range. Then tell whether the function is discrete or continuous.

A low-flow shower head releases 1.8 gallons of water per minute. The function V(x) gives the volume of water released after x minutes.

f(x) = 1.8x

x y0 0.5 .92.5 4.55 9

Domain: x 0Range: y 0

Continuous

graph