1.3 Graphs of Functions 2015 Digital Lesson. Warm-up/ Quiz Practice Copyright © by Houghton Mifflin...

1.3Graphs of Functions

2015

Digital Lesson

Warm-up/ Quiz Practice

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

a) State the domain of: 5 3f x x

1b) State the domain of:

4f x

x

Precalculus 1.3 Graphs of Functions Objectives

• Determine intervals on which functions are increasing, decreasing, or constant.

• Determine relative maximum and relative minimum values of functions.

• Identify and graph piecewise-defined functions.

• Identify even and odd functions.


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

Copyright © Houghton Mifflin Company. All rights reserved. Digital Figures, 1–8

Illustration of Domain and 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)


x

y

4

-4

Vertical Line Test

A relation is a function if no vertical line intersects its graph in more than one point.

This graph does not pass the vertical line test. It is not a function.

This graph passes the vertical line test. It is a function.

y = x – 1x = | y – 2|

x

y

4

-4


• decreasing on an interval if, as x values increase, y values decrease.

• constant on an interval if, as x values increase, y values remain constant.

The graph of y = f (x):

• increases on (– ∞, –2),

• decreases on (–2, 1.5),

• increases on (1.5, ∞).

A function f is:

• increasing on an interval if, as x values increase, y values increase.

(1.5, – 2)

x

y(–2, 3)

–2

2

5 - 12Copyright © 2010 Pearson Education, Inc.

Increasing, Decreasing, and EndpointsIncreasing, Decreasing, and Endpoints

The concepts of increasing and decreasing apply The concepts of increasing and decreasing apply only to intervals of the real number line and only to intervals of the real number line and NOT to individual points.NOT to individual points.

Do NOT say that the Do NOT say that the function function ff both both increases and increases and decreases at the point decreases at the point (0, 0).(0, 0).

Decreasing: (–∞, 0)Decreasing: (–∞, 0)Increasing: (0, ∞)Increasing: (0, ∞)

5 - 13Copyright © 2010 Pearson Education, Inc.

Ex 2: Determining intervals of increase or decreaseEx 2: Determining intervals of increase or decrease

Use the graph ofUse the graph of

Decreasing:Decreasing:

and

interval notation to identify where f is increasing or decreasing.

f x 4x 1

3x3

SolutionSolution

( , 2) (2, )

Increasing:Increasing: ( 2,2)

You try: find the intervals on which the function increases, decreases, or is constant. Remember that intervals are x-values.


increases on (– 5, –3), (-1,1), (4, 5)

decreases on (– ∞, -5), (1,4)

Constant on (–3, -1), (5, ∞)


A function value f(a) is called a relative minimum of f if it is less than all other function values (y values) nearby.

x

y

A function value f(a) is called a relative maximum of f if it is greater than all other function values (y values) nearby.

Relative minimum

Relative maximum


Section 1.3, Figure 1.24, Illustration of Definition of Relative Minimum and Relative Maximum, pg. 33


x

y

4

-4

A piecewise-defined function is composed of two or more functions.

f(x) =3 + x, x < 0

x2 + 1, x 0 Use when the value of x is less than 0.

Use when the value of x is greater or equal to 0.

(0 is not included.)

open circle

(0 is included.)

closed circle

Let’s try one together

2

2, 1

4, 1

x xf x

x x

Sketch the graph of

You try:

Sketch the graph of

2 3, 1

4, 1

x xf x

x x

Even and Odd Functions

• Some functions are even, some are odd, and some are neither.

• You can determine if a function is even, odd, or neither with an algebraic test, or by looking at its graph.



Symmetry of Graphs


A function f is even if for each x in the domain of f, f (– x) = f (x).

x

yf (x) = x2

f (– x) = (– x)2 = x2 = f (x)

f (x) = x2 is an even function.

For polynomials, every term in an even function will have an even power. Constants are ok, they are zero power.

Symmetric with respect to the y-axis.


A function f is odd if for each x in the domain of f, f (– x) = – f (x).

x

y

f (x) = x3

f (– x) = (– x)3 = –x3 = – f (x)

f (x) = x3 is an odd function.For polynomials, every term in an odd function will have an odd power, Constants are not ok.

Symmetric with respect to the origin.

Example

Are the following functions even, odd, or neither?


g x x x( ) 3

h x x( ) 2 1

f x x( ) 3 1

odd

even

neither

Homework

• Pg. 38: 11-25 odd, 29, 45, 47, 49-55 odd, 109


HWQ

• State intervals of increase and decrease on the graph of .


3 26 15f x x x

1.3 Graphs of Functions 2015 Digital Lesson. Warm-up/ Quiz Practice Copyright © by Houghton Mifflin...

Documents

Transcript of 1.3 Graphs of Functions 2015 Digital Lesson. Warm-up/ Quiz Practice Copyright © by Houghton Mifflin...