This document discusses different types of piecewise functions including: - Piecewise functions which use separate equations depending on the domain values. For example, one equation for when x ≤ 1 and another for when x > 1. - An example problem evaluating a piecewise function at different x values and determining which equation to use. - Graphing a piecewise function involves using one equation's graph for values below a breaking point, and another equation's graph for values at or above the breaking point. - Greatest integer functions which return the greatest integer less than or equal to the input value. An example graph of y = [x] + 2 is shown and evaluated in a table.

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, 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.

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 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)

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 ] 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. 2 4 6 – 2 – 4 – 6 x 2 4 6 – 2 – 4 – 6 y y =[ x ]

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 -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

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