1. Piecewise Functions Objective: Students will be able to graph, write and evaluate piecewise functions.

2. Relation – a mapping of input and output values Function - a relation that has a unique output for each input ( every x has a unique y ) vertical line test – a relation is a function if and only if no vertical line intersects the graph of the relation at more than one point. Domain – the input values, x values, independent variable Range – the output values, y values, dependent variable

3. 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.

4. Piecewise Functions A function made up of a combination of equations, each corresponding to a part of the domain.

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

6. 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

7. 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)

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

9. Graph: Point of Discontinuity

10. Step Functions

12. Graph :

14. Graphing a Piecewise Function Graph each part of the function individually but put them on the same graph Graph this function: -x + 3, if x ≥ 1

15. Evaluating a Piecewise Function Evaluate f(x) when: x = 0, 2 and 4 f(x) = x + 2, if x <2 2x + 1, if x ≥ 2

16. Snowstorm During a ten hour snowstorm it snows at a rate of 1 inch per hour for the first 3 hours, at a rate of 2 inches per hour for the next six hours and 1 inch per hour for the final hour. Write and graph a piecewise function that gives the depth of snow during the snowstorm. How many inches of snow accumulate from the storm?

17. The Absolute Value Function The Absolute Absolute An Absolute Value FunctionAn Absolute Value Function is a famous Piecewise Function. It has two pieces: below zero: -x from 0 onwards: x f(x) = |x|

18. Step Functions the graphs resemble a set of stair steps The greatest integer function is a step function For every real number x, g(x) is the greatest integer less than or equal to x

19. Writing a piecewise function

20. Using a step function A parking garage charges $3 for the first hour and $8 for a maximum of twelve hours ($3 for the first hour and $8 for hours 2-12) Write and graph a piecewise function for the parking charges.

21. You have a summer job that pays time and a half for overtime. If you work more than 40 hours per week, your hourly wage for the extra hours is 1.5 times your normal hourly wage of $7. Write and graph a piecewise function that gives your weekly pay P in terms of the number of hours, h, you work. How much will you get paid if you work 45 hours?

22. 14. You have a summer job that pays time and a half for overtime. If you work more than 40 hours per week, your hourly wage for the extra hours is 1 1/2 times your normal hourly wage of $10. a.) Write a piecewise function that gives your weekly pay P in terms of the number of hours, h, you work. b.) How much will you get paid if you work 46 hours?