1. Algebra II Piecewise Functions Edited by Mrs. Harlow

2. 2-6: Special Functions Constant Identity Absolute Value Step/Greatest Integer Piecewise

3. Constant Function: A linear function in the form y = b. y = 3

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

5. Up to now, weve 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.

6. One equation gives the value of f(x) when x 1 And the other when x>1

7. Evaluate f(x) when x=0, x=2, x=4 First you have to figure out which equation to use You NEVER use both X=0 This one fits into the top equation So: 0+2=2 f(0)=2 X=2 This one fits here So: 2(2) + 1 = 5 f(2) = 5 X=4 This one fits here So: 2(4) + 1 = 9 f(4) = 9

8. Graph: For all xs < 1, use the top graph (to the left of 1) For all xs 1, use the bottom graph (to the right of 1)

9. x=1 is the breaking point of the graph. To the left is the top equation. To the right is the bottom equation.

10. Graph:

11. Greatest Integer Function: A function in the form y = [x] 246–2–4–6x 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.

12. Greatest Integer Function: A function inthe form y = [x] 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 0 1 246–2–4–6x 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

13. Step Functions

15. Graph :

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

18. Assignment