graphs of functions digital lesson. copyright © by houghton mifflin company, inc. all rights...

Graphs of Functions

Digital Lesson

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

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y

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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)


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y

4

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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|

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• 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

Increasing/Decreasing Intervals

Calculator Mode Screen

Keep your calculator in degree mode until we get to trigonometry.

Standard Window ScreenThis is where you change

the size of your graph.Xmin is the leftmost point on

your graph.Xmax is the rightmost point

on your graph.Xscl is the scale you are using

on your x-axis.Ymin is the lowest point on

your graph.Ymax is the highest point on

your graph.Yscl is the scale that you are

using on your y-axis.

The Math Menu

Full, Horizontal, G-T Options

This alteration is made in the MODE options.

Grids

FORMAT is located in the text above the ZOOM button located at the top center of your calculator.


A function value f(a) is called a relative minimum of f if there is an interval (x1, x2) that contains a such that

x1 < x < x2 implies f(a) f(x).

x

y

A function value f(a) is called a relative maximum of f if there is an interval (x1, x2) that contains a such that

x1 < x < x2 implies f(a) f(x).

Relative minimum

Relative maximum

Relative Max and Min

Relative Max/Min


Graphing Utility: Approximate the relative minimum of the function f(x) = 3x2 – 2x – 1.

– 6

– 6

6

6


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y

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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

Piecewise Functions

TEST is located in the text right above the MATH button on the left side of the calculator.

Piecewise Functions

Greatest Integer Function

Greatest Integer Functions


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.

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.

Symmetric with respect to the origin.

Even/Odd Functions


Even/Odd Function


graphs of functions digital lesson. copyright © by houghton mifflin company, inc. all rights...

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