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