Algebra IIAlgebra IIPiecewise Functions Piecewise Functions

Edited by Mrs. HarlowEdited by Mrs. Harlow

2-6: Special Functions2-6: Special Functions

• Constant• Identity• Absolute Value• Step/Greatest Integer • Piecewise

Constant Function: Constant Function: A linear function in the form A linear function in the form y y = = b.b.

2 4 6–2–4–6 x

2

4

6

–2

–4

–6

y

y = 3

Identity Function: Identity Function: A linear function in the form A linear function in the form y y = = x.x.

2 4 6–2–4–6 x

2

4

6

–2

–4

–6

y

y=x

• Up to now, we’ve been looking at functions represented by a single equation.

• In real life, however, functions are represented by a combination of equations, each corresponding to a part of the domain.

• These are called piecewise functions.

2 1,if 1

3 1,if 1

x xf x

x x

•One equation gives the value of f(x) when x ≤ 1•And the other when x>1

Evaluate f(x) when x=0, x=2, x=4Evaluate f(x) when x=0, x=2, x=4

2,if 2( )

2 1,if 2

x xf x

x x

•First you have to figure out which equation to use•You NEVER use both

X=0This one fitsinto the top equation

So:0+2=2f(0)=2

X=2This one fits hereSo:2(2) + 1 = 5f(2) = 5

X=4

This one fits hereSo:2(4) + 1 = 9f(4) = 9

Graph:Graph:31

2 2 , if 1( )

3, if 1

x xf x

x x

•For all x’s < 1, use the top graph (to the left of 1)•For all x’s ≥ 1, use the bottom graph (to the right of 1)

312 2 , if 1

( )3,if 1

x xf x

x x

x=1 is the breakingpoint of the graph.

To the left is the topequation.

To the right is thebottom equation.

Graph:Graph:

2 2,if 2

3 3( )1, if 2

x xf x

x x

Greatest Integer Function: A function in the form Greatest Integer Function: A function in the form y y = [= [xx]]

2 4 6–2–4–6 x

2

4

6

–2

–4

–6

y

y=[x]

Note: [x] means the greatest integer less than or equal to x. For example, the largest integer less than or equal to -3.5 is -4.

Greatest Integer Function: A function inthe form Greatest Integer Function: A function inthe form y y = [= [xx]]

Graph y= [x] + 2 by completing the t-table:

x y -3 -2.75 -2.5 -2.25 -2 -1.75 -1.5 -1.25 -1 0 1

2 4 6–2–4–6 x

2

4

6

–2

–4

–6

y

x y -3 y= [-3]+2=-1 -2.75 y= [-2.75]+2=-1 -2.5 y= [-2.5]+2=-1 -2.25 y= [-2.25]+2=-1 -2 y= [-2]+2 =0 -1.75 y= [-1.75]+2=0 -1.5 y= [-1.5]+2=0 -1.25 y= [-1.25]+2=0 -1 y= [-1]+2=1 0 y= [0]+2=2 1 y= [1]+2=3

Step FunctionsStep Functions

1,if 0 12,if 1 2( )3, if 2 34,if 3 4

xxf xxx

1,if 0 12,if1 2( )3, if 2 34,if 3 4

xxf xxx

Graph :Graph :

1,if 4 32,if 3 2( )3, if 2 14,if 1 0

xxf xxx

Labor costs at the Fix-It Auto Repair Shop are $60 per hour or any fraction thereof. Draw a graph that represents this situation.

Assignment

1. Graph ( ) 3 2

(the greatest integer function has been modified...)

5, if 2

2. ( ) 2 1, if 2 2

2 9, if 2

2 1, if 33. Graph: ( )

8, if 3

f x x

x x

f x x x

x x

x xf x

x x