1. Piecewise Functions Piecewise-Defined Functions

2. 7/9/2013 Piecewise Defined Functions 2 Piecewise-Defined Functions Driving to Aunt Berthas House Your family drives to Aunt Berthas house for Thanksgiving dinner, a distance of 100 miles You average 50 mph each way, visit for 5 hours and return home Define and graph D(t) as distance from home at time t

3. 7/9/2013 Piecewise Defined Functions 3 Piecewise-Defined Functions Driving to Aunt Berthas House Define and graph D(t) as distance from home at time t D(t) = 50t, if 0 t < 2 100, if 2 t < 7 –50t + 450, if 7 t 9 Questions: What is the domain of D ? What is the range of D ? Is function D(t) continuous ?

4. 7/9/2013 Piecewise Defined Functions 4 Piecewise-Defined Functions Driving to Aunt Berthas House Slope = D t = –100 miles 2 hours = –50 hr mi D(t) t 100 50 0279 D(t) = 50t, if 0 t < 2 100, if 2 t < 7 –50t + 450, if 7 t 9 = 100 miles 2 hours = 50 hr mi Slope = D t Domain Range

5. 7/9/2013 Piecewise Defined Functions 5 Piecewise-Defined Functions The Postal Function In 2008 first-class letter postage P(x) for letters of weight x ounces was assigned as follows: P(x) =.42.59.76.93 1.34, if 0 x 1, if 1 < x 2, if 2 < x 3, if 3 < x 3.5, if 3.5 < x 4 1.51, if 4 < x 5 Question: Graph ?

6. 7/9/2013 Piecewise Defined Functions 6 Piecewise-Defined Functions The Postal Function 012345 3.5.42.59.76.93 1.34 1.51 P(x) x Question: Graph ? A step function Domain ? Range ? Question: Is the function P(x) continuous ?

7. 7/9/2013 Piecewise Defined Functions 7 The Greatest-Integer Function x nxnx -3 -2 1 2 3 -2 -3 3 2 1 Piecewise-Defined Functions n x is the greatest integer x For each non-integer x the value of n x is the largest integer to the left of x Between consecutive integers is constant x nxnx x

8. 7/9/2013 Piecewise Defined Functions 8 The Greatest-Integer Function x nxnx -3 -2 1 2 3 -2 -3 3 2 1 Piecewise-Defined Functions n x is the greatest integer x For integer a, if a < x < a + 1, then Questions : Domain Range ? ? nxnx x nxnx x

9. 7/9/2013 Piecewise Defined Functions 9 The Greatest-Integer Function Piecewise-Defined Functions n x is the greatest integer x Questions : Where ? Is continuous ? x nxnx -3 -2 1 2 3 -2 -3 3 2 1 nxnx x = k k x k What is n k for some integer k? =

10. 7/9/2013 Piecewise Defined Functions 10 x A(x) Years Since Birth Current Age 1 2 3 4 5 6 2 1 5 4 3 6 Piecewise-Defined Functions How Old Are You ? In your first year, what is your age ? When does your age change ? How old are you during your third year ? 0 years At 1 year 2 years

11. 7/9/2013 Piecewise Defined Functions 11 Piecewise-Defined Functions How Old Are You ? In your first year, what is your age ? In general, if you are n years old, in what year of your life are you ? 0 years Year n+1 x A(x) Years Since Birth Current Age 1 2 3 4 5 6 2 1 5 4 3 6

12. 7/9/2013 Piecewise Defined Functions 12 x F(x) –1 1 The Salt-and-Pepper Function Piecewise-Defined Functions 1 Question: the domain of F(x) ? the range of F(x) ? F(x) = –1, if x rational, if x irrational Slope not defined Nowhere-continuous function F(0), F(-6), F(), F( ), F 2 ? ( ) What are

13. 7/9/2013 Piecewise Defined Functions 13 The Absolute Value Function Definition The absolute value of a number a, written a, is defined by The absolute value function is for all x Piecewise-Defined Functions a = a –a, if a 0, if a < 0 F(x) x 1 2 –112 –2 F(x) = | x | ( 2, 2 ) || ( -2, -2 ) ||

14. 7/9/2013 Piecewise Defined Functions 14 The Absolute Value Function The absolute value function is for all x Piecewise-Defined Functions F(x) = | x | F(x) x 1 2 –112 –2 ( 2, 2 ) || ( -2, -2 ) || Question: What is the domain of F(x) ? What is the range of F(x) ? Is F(x) continuous ? F(0), F(-6), F(), F( ), F 2 ? ( ) What are : YES

15. 7/9/2013 Piecewise Defined Functions 15 Absolute Value Function Variations Example 1 : Vertex: f(-1) = 0 Piecewise-Defined Functions Questions: Domain of f(x) ? Range of f(x) ? Is f(x) continuous ? What are: f(0), f(-6), f(), f(- ) ? … for all x f(x) x 1 2 –112 –2 ( 1, 2 ) || ( -3, -2 ) || f(x) = x + 1 | |

16. 7/9/2013 Piecewise Defined Functions 16 Absolute Value Function Variations Example 2 : Vertex: f(1) = 0 Piecewise-Defined Functions … for all x f(x) = x – 1 | | f(x) x 1 2 –112 –2 ( 3, 2 ) || ( -1, -2 ) || Questions: Domain of f(x) ? Range of f(x) ? Is f(x) continuous ? What are: f(0), f(-6), f(), f(- ) ?

17. 7/9/2013 Piecewise Defined Functions 17 Think about it !