Algebra II Algebra II Piecewise Functions Piecewise Functions Edited by Mrs. Harlow Edited by Mrs. Harlow
May 24, 2015
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