Intermediate Algebra (11th edition)

Nonlinear Systems Nonlinear Functions, Conic Sections, and

OBJECTIVES OBJECTIVE 1 Recognize the graphs of the elementary functions defined

by , and and graph their translations. Earlier, we introduced the

squaring functiondefined by. Another elementary function, defined by

,is the absolute value function.This function pairs each real number

with its absolute value. Its graph is shown in FIGURE 1.

ƒ 1 x 2 |x|

ƒ 1 x 2 =x^2

|x|,^1 x 2 x,

Additional Graphs of Functions

11.1

1 Recognize the

graphs of the

elementary

functions defined

by , and

and graph their

translations.

2 Recognize and

graph step

functions.

|x|,^1 x 2 x,

y

0

1

2

3

+-

+-

+-

0

1

2

3

x

(0, 0)

x

y

2

• 2


• 2

2

f f (x) x

FIGURE 1

The reciprocal function,defined by and introduced in Section 7.4,is

a rational function. Its graph is shown in FIGURE 2. Since xcan never equal 0, as

xgets closer and closer to 0, approaches either or Also, can never equal 0,

and as xapproaches or approaches 0. The axes are called asymptotesfor the

function.

1

- x

q q,

1

- x

(^1) q q.
x

ƒ 1 x 2 ^1 x

2

0

1

x x

y y

• 1


• 3


• 2


• 2


• 3


• 1 – 1
  1

1

2

(^133)
(^313)
2
11
1
2

x x y y

-^12
-^13
-^13
-^12

x y

f f (x)^1 x

HorizontalHorizontal

asymptoteasymptote

y 0

VerticalVertical

asymptoteasymptote

x 0

FIGURE 2

The square root function,defined by and introduced in Section 8.1,

is shown in FIGURE 3.

ƒ 1 x 2  2 x

y

0

1

4

0

1

2

x

(0, 0)

x

y

2

• 2

4

f (x) x

FIGURE 3

xand y

are always

nonnegative.

Reciprocal function

Domain:

Range: 1 - q, 0 2 ́ 1 0, q 2

1 - q, 0 2 ́ 1 0, q 2

ƒ 1 x 2 

1

x

Square root function

Domain:

Range: 3 0, q 2

3 0, q 2

ƒ (x ) 2 x

Absolute value function

Domain:

Range: 3 0, q 2

1 - q, q 2

xcan take on ƒ^1 x^2 |x|

any real number

value.

yis always

nonnegative.