Section 3.4 Basic FunctionsConstant Function ( )f x c

Linear FunctionIdentity

( )f x x

Section 3.4 Basic FunctionsSquare FunctionQuadratic

2( )f x x

Cubic Function 3( )f x x

Section 3.4 Basic Functions

Absolute Value Function

( )f x x

Cube Root Function

3( )f x x

Section 3.4 Basic FunctionsRational Function

1( )f x

x

Square Root Function

2( )f x x

Section 3.4 Basic FunctionsGreatest Integer Function xxxf int

Section 3.4 Basic FunctionsContinuous Functions

Is a function where the graph has no gaps, holes, or breaks…it can be drawn without stopping and lifting your pencil.

Discontinuous Functions

Is a function where the graph has gaps, holes, or breaks…it can not be drawn without stopping and lifting your pencil.

Piecewise-Function

( )f x {for

for

for

Find the following… 2f 3f 1f

Find the inequality that is true for the value of x and plug the value into the function.

1

1

13

x

x

x

1

2

12

2

x

x

21 f 8133 2 f 51222 f

2. What is the domain of the function?Use the individual domain restrictions to find the entire domain. ,3D

Piecewise-Function

( )f x {for

for

for 1

1

13

x

x

x

1

2

12

2

x

x

3. Find all intercepts.Y-intercept is when x = 0 or f(0). 1102 The y-intercept is at ( 0, -1 ).

x-intercepts are when y = f(x) = 0. Take each individual function and set it equal to zero.

012 x 02 012 x12 x

2

1x

Not possible, ½ is not in the domain.

Not true. 12 x1x

x = 1 is ok, but -1 violates domain.

The x-intercept is at ( 1, 0 ).

Piecewise-Function

( )f x {for

for

for 1

1

13

x

x

x

1

2

12

2

x

x

4. Graph the function.Test all endpoints given in the domain restrictions.Color coordinate the functions.

Test the -3 and -1 for the first function.-2(-3) + 1 and -2(-1) + 1 6 + 1 = 7 2 + 1 = 3 ( -3, 7 ) ( -1, 3 )

Closed point because of equal to line.

Open point because of no equal to line.

Test the -1 for the second function. This is a constant function with only x = -1, yields the point ( -1, 2 ).

Test the -1 for the third function, but it will be an open point. This is a quadratic function, so we will test points, 0, 1, 2, etc.

(-1)2 – 1 = 0( -1, 0 ) (0)2 – 1 = 0

( 0, -1 )

(1)2 – 1 = 0( 1, 0 ) (2)2 – 1 = 0

( 2, 3 )

(3)2 – 1 = 0( 3, 8 )

Piecewise-Function

( )f x {for

for

for 1

1

13

x

x

x

1

2

12

2

x

x

5. Use the graph to determine the range.

Use your pencil as a horizontal line. Start at the lowest point on the graph and slide your pencil up. Identify all the y – coordinates that your touching.

1,

6. Is f continuous on its domain? No

Piecewise-Function

( )f x {

1

12

54 2

x

x

x

for

for

for

1

12

2

x

x

x

)2(f

)5(f

)1(f

1. What is f(-2), f(5), and f(-1 )?

15

542 2 52 2 154

121

24

211

2. What is the domain of the function? ,:D

Piecewise-Function

( )f x {

1

12

54 2

x

x

x

for

for

for

1

12

2

x

x

x

4. Graph the function.

1546

4545

5544

4543

1542

54

2

2

2

2

2

2

y

y

y

y

y

xy

215

112

011

1

y

y

y

xy

6

5

4

3

2

x

4121

3120

2121

1122

12

y

y

y

y

xy

1

0

1

2

x

5

2

1

x

Piecewise-Function

( )f x {

1

12

54 2

x

x

x

for

for

for

1

12

2

x

x

x

3. Find all intercepts.

x intercepts

54

54

54

54

054

2

2

2

x

x

x

x

x 0,54

y intercept

3,0 5. Find the Range.

,

6. Is f continuous on its domain? No