1. Piecewise Functions

2. Warm up:
Write down all of the information you know about a piecewise function.
Write down what is still confusing to you about piecewise functions
Talk to your group members – Write down what you know as a group

3. What is a Piecewise function?
A function which is defined by multiple sub-functions, each sub- function applying to a certain interval of the main function's domain (a sub-domain).

4. What is Domain?
The set of values of the independent variable(s) for which a function or relation is defined. Typically, this is the set of x-values that give rise to real y-values.
Note: Usually domain means domain of definition, but sometimes domain refers to a restricted domain.

5. What is Range?
The complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain.
In plain English, the definition means: The range is the resulting y-values we get after substituting all the possible x-values

6. Why are Domain and Range important to Piecewise functions?
We need to have a specified domain (and range) to specify where each piece of the function starts and finishes, so a new one can begin.

7. What makes our inequality signs different?
Symbols mean?
When graphing?
<
> ≤ ≥
<
> ≤ ≥

8. Graphing a piecewise function
In order to graph a piecewise function, we must graph ALL of the individual parts, within their restricted domains
A piecewise can have two parts or a larger number of parts

9. Example 1:
𝑓 𝑥 = 2, 𝑥<0 &5, 𝑥≥0
Which has an open circle? Closed circle?
Where does one piece end and the next begin?
What is the range of each function?

10. Example 2: 𝑓 𝑥 = −𝑥+2, 𝑥<2 &𝑥−2, 𝑥≥2
𝑓 𝑥 = −𝑥+2, 𝑥<2 &𝑥−2, 𝑥≥2
Which has an open circle? Closed circle?
Where does one piece end and the next begin?
What is the range of each function?

11. Example 3: 𝑓 𝑥 = −3𝑥+2, 𝑥≤2 & 1 2 𝑥−4, 𝑥>2
𝑓 𝑥 = −3𝑥+2, 𝑥≤2 & 1 2 𝑥−4, 𝑥>2
Which has an open circle? Closed circle?
Where does one piece end and the next begin?
What is the range of each function?

12. You try: 𝑓 𝑥 = −𝑥, 𝑥<0 𝑥, 𝑥≥0
𝑓 𝑥 = −𝑥, 𝑥<0 𝑥, 𝑥≥0
Which has an open circle? Closed circle?
Where does one piece end and the next begin?
What is the range of each function?

13. You try: 𝑓 𝑥 = 2𝑥, 𝑥<−1 1 2 𝑥, 𝑥≥2
𝑓 𝑥 = 2𝑥, 𝑥<− 𝑥, 𝑥≥2
Which has an open circle? Closed circle?
Where does one piece end and the next begin?
What is the range of each function?

14. Piecewise Day 2

15. Warm up: Try 𝑓 𝑥 = −3𝑥+12, 𝑥≤−3 𝑥−4, −3<𝑥<3 &3𝑥−12, 𝑥≥3
𝑓 𝑥 = −3𝑥+12, 𝑥≤−3 𝑥−4, −3<𝑥<3 &3𝑥−12, 𝑥≥3
Which has an open circle? Closed circle?
Where does one piece end and the next begin?
What is the range of each function?

16. Evaluating a Piecewise Function at various values
How do we evaluate a function?
For piecewise functions, we must evaluate within a specific parameter – that means you need to be careful about which sub function you use to evaluate.

17. Example 1:
𝑓 𝑥 = 2, 𝑥<0 &5, 𝑥≥0
𝑓 2
𝑓 −3
𝑓(0)

18. Example 2:
𝑓 𝑥 = −𝑥+2, 𝑥<2 &𝑥−2, 𝑥≥2
𝑓 2
𝑓 −3
𝑓(0)

19. Example 3:
𝑓 𝑥 = −3𝑥+2, 𝑥≤2 & 1 2 𝑥−4, 𝑥>2
𝑓 2
𝑓 −3
𝑓(0)

20. Example 4: 𝑓 𝑥 = −3𝑥+12, 𝑥≤−3 𝑥−4, −3<𝑥<3 &3𝑥−12, 𝑥≥3 𝑓 2 𝑓 −3
𝑓 𝑥 = −3𝑥+12, 𝑥≤−3 𝑥−4, −3<𝑥<3 &3𝑥−12, 𝑥≥3
𝑓 2
𝑓 −3
𝑓(0)

21. You try:
𝑓 𝑥 = −𝑥, 𝑥<0 𝑥, 𝑥≥0
𝑓 2
𝑓 −3
𝑓(0)

22. You try:
𝑓 𝑥 = 2𝑥, 𝑥<− 𝑥, 𝑥≥2
𝑓 2
𝑓 −3
𝑓(0)

23. Writing the function of a Piecewise.
How do we write the equation of a line?
For piecewise functions, we must write equations for each different piece, then give the parameters.
Step 1 : Find the parameters
Step 2 : Find the equation of one line at a time
Step 3 : put all parts into the form for a piecewise

24. Example 1: 1) Identify the parameters:
2) What are the individual lines?
3) 𝑓 𝑥 = 𝑖𝑓 ______________ 𝑖𝑓 ______________

25. Example 2: 1) Identify the parameters:
2) What are the individual lines?
3) 𝑓 𝑥 = 𝑖𝑓 ______________ 𝑖𝑓 ______________

26. Example 3: 1) Identify the parameters:
2) What are the individual lines?
3) 𝑓 𝑥 = 𝑖𝑓 ______________ 𝑖𝑓 ______________ 𝑖𝑓 ______________

27. You try: 1) Identify the parameters: 2) What are the individual lines?
3)𝑓 𝑥 = 𝑖𝑓 ______________ 𝑖𝑓 ______________

28. You try: 1) Identify the parameters: 2) What are the individual lines?
3) 𝑓 𝑥 = 𝑖𝑓 ______________ 𝑖𝑓 ______________ 𝑖𝑓 ______________

29. Homework
Use homework packet – complete first page