1. 2.6: Special Functions
Algebra 2

2. Piecewise function: A function written using two or more expressions each corresponding to a part of the domain.
Why we use them: sometimes the output of functions depends on the values being put into them

4. Evaluating a piecewise function: use the domain to determine which function to plug the value into
Example 1: Evaluate f(x) when
a) x = 0 b) x = 3 c) x = 6

5. Example 2: Evaluate f(x) when a) x = -3 and b) x = -1

6. Graphing a Piecewise Function
1. Make a table of 3 values for each function. The x values in the table should follow from the domain 2. Determine if the initial point of each piece of the graph should be an open circle or a closed circle 3. Plot the points from your tables and connect to make rays

7. Example 3: Graph the piecewise function and then identify domain and range.

10. Write the piecewise-defined function shown in the graph.

12. Step Functions: a graph consisting of horizontal line segments resembling steps
Graphing Step Functions:
1. Pay attention to whether circles should be open or closed
2. Make horizontal segments at the appropriate y value for the length of the domain for each function.

14. One psychologist charges for counseling sessions at the rate of $85 per hour or any fraction thereof. Draw a graph that represents this situation.

15. Shipping costs $6 on purchases up to $50, $8 on purchases over $50 up to $100, and $10 on purchases over $100 up to $200. Write a step function for these charges. Give the domain and range.

16. The greatest integer function of a real number x is represented by [x]. 
For all real numbers x, the greatest integer function returns the largest integer less than or equal to x.  In other words, the greatest integer function rounds down a real number to the nearest integer.

17. Examples
6.2 =
5.9 =
8 =
−2.4 =

18. The graph
One endpoint in each step is closed (black dot) to indicate that the point is a part of the graph and the other endpoint is open (open circle) to indicate that the points is NOT a part of the graph.

19. Graph𝑓 𝑥 = 𝑥 +1

20. Graph𝑓 𝑥 =2 𝑥 −3

21. Graph𝑓 𝑥 = 3𝑥−2