1 5 graphs of functions

Graphs of Functions

Digital Lesson

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

The graph of a function f is the collection of ordered pairs (x, f(x)) where x is in the domain of f.

x

y

4

-4

(2, –2) is on the graph of f(x) = (x – 1)2 – 3.

(2, –2)

f(2) = (2 – 1)2 – 3 = 12 – 3 = – 2


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)


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, 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


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


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

– 6

– 6

6

6

– 0.86– 4.79

– 1.79

2.14

0.58 0.76

-3.24

-3.43

Zoom In:

Zoom In: The approximate minimum is

(0.67, –3.33).


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


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.

1 5 graphs of functions

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