Intermediate Algebra (11th edition)

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638 CHAPTER 11 Nonlinear Functions, Conic Sections, and Nonlinear Systems

y

(^3) x
–2
0
(–1, –4) f(x) = √x + 1 – 4
FIGURE 6
xy
3 - 2
0 - 3

• 1 - 4

Compare this table

of values to that

with FIGURE 3.

Domain:

Range: 3 - 4, q 2

3 - 1, q 2

OBJECTIVE 2 Recognize and graph step functions. The greatest integer

function is defined as follows.

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3.

domain: ; range: 1 - q, q 2 3 - 3, q 2

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EXERCISE 3
Graph
Give the domain and range.

ƒ 1 x 2 =|x+ 1 |-3.

x

y

–4

–3

–1 2

0

f(x) = x + 1 – 3

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EXERCISE 4
Evaluate each expression.

(a) (b)

(c) 3.5 (d) -4.1

 5  - 6 

4. (a) 5 (b)- 6 (c) 3 (d)- 5

The greatest integer function,written pairs every real number x

with the greatest integer less than or equal to x.

ƒ 1 x 2 = x,

ƒ 1 x 2 x

Finding the Greatest Integer

Evaluate each expression.

(a) (b) (c)

(d) The greatest integer less than or equal to7.45 is 7.

(e) -2.6 =- 3

7.45= 7

 8  = 8 - 1  =- 1  0 = 0

EXAMPLE 4

• 3 –2 –1 0

–2.6

Think of a number line with graphed on it. Since is to the left of(and

is, therefore, less than) the greatest integer less than or equal to is

not-2.

- 2.6, -2.6 -3,

- 2.6 - 3

Graphing the Greatest Integer Function

Graph Give the domain and range.

For if then

if then

if then

if then

if then and so on.

Thus, the graph, as shown in FIGURE 7on the next page, consists of a series of

horizontal line segments. In each one, the left endpoint is included and the right

endpoint is excluded. These segments continue infinitely following this pattern to

the left and right. The appearance of the graph is the reason that this function is

called a step function.

3 ...x 6 4, x = 3,

2 ...x 6 3, x = 2;

1 ...x 6 2, x = 1;

0 ...x 6 1, x = 0;

x, - 1 ...x 6 0, x =-1;

ƒ 1 x 2 = x.

EXAMPLE 5

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