1. Ch 4-3 Piecewise and Stepwise Functions C.N.Colon ALGEBRA-HP SBHS

2. DO NOW Solve and graph the following equations. 1. 2.

3. QUESTION: How are absolute value equations similar to piecewise functions? Today’s Question: How do we graph piecewise functions?

4. 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.

5. A piecewise function is a function that is a combination of one or more functions. The rule for a piecewise function is different for different parts, or pieces, of the domain For instance, movie ticket prices are often different for different age groups. So the function for movie ticket prices would assign a different value (ticket price) for each domain interval (age group).

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: Point of Discontinuity

11. A piecewise function that is constant for each interval of its domain is called a step function.

12. Step Function

14. Graph :

16. Special Step Functions Two particular kinds of step functions are called ceiling functions ( f (x)= and floor functions ( f (x)= ).

17. Special Step Functions In a ceiling function, all non-integers are rounded up to the nearest integer. An example of a ceiling function is when a phone service company charges by the number of minutes used and always rounds up to the nearest integer of minutes. ( f (x)=

18. Special Step Functions In a floor function, all non-integers are rounded down to the nearest integer. The way we usually count our age is an example of a floor function since we round our age down to the nearest year and do not add a year to our age until we have passed our birthday. The floor function is the same thing as the greatest integer function which can be written as f (x)=[x].

19. Evaluating Piecewise Functions Evaluating piecewise functions is like evaluating functions that you are already familiar with. f(x) = x 2 + 1, x  0 x – 1, x  0 Let’s calculate f(2). You are being asked to find y when x = 2. Since 2 is  0, you will only substitute into the second part of the function. f(2) = 2 – 1 = 1 EXTRA Examples

20. f(x) = x 2 + 1, x  0 x – 1, x  0 Let’s calculate f(-2). You are being asked to find y when x = -2. Since -2 is  0, you will only substitute into the first part of the function. f(-2) = (-2) 2 + 1 = 5

21. Your turn: f(x) = 2x + 1, x  0 2x + 2, x  0 Evaluate the following: f(-2) = -3? f(0) = 2? f(5) = 12 ? f(1) = 4? No answers will be given here….

22. One more: f(x) = 3x - 2, x  -2 -x, -2  x  1 x 2 – 7x, x  1 Evaluate the following: f(-2) = 2? f(-4) = -14? f(3) = -12 ? f(1) = -6 ? No answers will be given here either….

23. Homework Textbook p. 158 # 1-3 and p. 161 #2-16 (e)