domain/range continuous/discontinuous increasing/decreasing constant

Domain/RangeContinuous/Discontinuous

Increasing/DecreasingConstant

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2

Objectives• I can find domain and range in Interval

Notation• I can identify increasing, decreasing, and

constant intervals of a function• I can tell if a function is continuous

Domain and Range

• The domain in any relation is the first coordinates from the ordered pairs. It is the Input!

• Domain = X -Values• The range in any relation is the second

coordinates from the ordered pairs. It is the Output!

• Range = Y- Values

Example 1: Domain/Range

• Given the following relation• {(2,3), (-4,8), (2,6), (7,-3)}• What is the Domain?• { -4, 2, 7}• **Notice they are listed least to greatest!! • No duplicates!!!• What is the Range?• {-3, 3, 6, 8}

x

y

4

-4

The domain of the function y = f (x) is the set of values of x for which a corresponding value of y exists.

The range of the function y = f (x) is the set of values of y which correspond to the values of x in the domain.

Domain

Range

x

y

– 1

1

Example: Find the domain and range of the function f (x) = from its graph.

The domain is [–3,∞).

The range is [0,∞).

3x

Range

Domain

(–3, 0)

Example 1Domain( , )

Range[ 3, )

Example 2

Domain( , )

Range( , 4]

Example 3

Domain[0, )

Range( , )


Functions

INCREASING

DECREASING

CONSTANT


• decreasing on an interval if, for any x1 and x2 in the interval, x1 < x2 implies f (x1) > f (x2),• constant on an interval if, for any x1 and x2 in the interval, f (x1) = f (x2).

The graph of y = f (x):

• increases on (– ∞, –3),

• decreases on (–3, 3),

• increases on (3, ∞).

A function f is:• increasing on an interval if, for any x1 and x2 in the interval, x1 < x2 implies f (x1) < f (x2),

(3, – 4)

x

y(–3, 6)

–2

2


2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

7

123456

8

-2-3-4-5-6-7

Look at the graph of the function shown on the interval (-6,-2)

This means x values between –6 and –2.

As you follow the graph of the function from x = -6 to x = -2, does the function value (remember that is the y value) increase, decrease, or remain constant (the same)?

It INCREASES so we say the function is increasing on the interval (-6, -2)

Can you see another interval where the function is increasing?

The function is also increasing on (4, 6)

x = 4x = -6 x = -2 x = 6

This is NOT an ordered

pair


2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

7

123456

8

-2-3-4-5-6-7

Can you see an interval where the function is decreasing?

The function is decreasing on the interval (-2, 4) since when you follow the graph between x = -2 and x = 4 the function value (y value) goes down.

Remember for an interval you list the x values that make the y values decrease. Always move from left to right on the graph (from smaller x values to larger x values). x = 4x = -2


2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

7

123456

8

-2-3-4-5-6-7

What is this function doing on the interval (-7, -2)?

It is

INCR

EASI

NGx = -2x = -7

What is this function doing on the interval (-2, 2)?

What is this function doing on the interval (2, 7)?

x = 2 x = 7

It is DECREASING

It is not increasing OR decreasing but remaining

constant

Continuous or Discontinuous??

• A function is continuous if it has an infinite domain and forms a smooth line or curve

• Simply put: It has NO BREAKS!!!

• You should be able to trace it with your pencil from left to right without picking up your pencil


Look at the following graphs and determine if they are Continuous

or Discontinuous Functions??



Homework

• WS 1-3• Quiz next class• Work on Parent Function Packet

domain/range continuous/discontinuous increasing/decreasing constant

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