1. 1.1 Functions and Change Friday, August 26, 2005

2. Some Objectives  Understand interplay between 4 ways of representing a function  Finding the domain and range of a function  Investigating even and odd functions  Working with piecewise defined functions

3. What is a Function?  Arise whenever one quantity depends on another

4. Some Definitions  A function f is a rule that assigns to each element in a set A exactly one element, called f(x), in a set B.  Usually consider functions for which the sets A and B are sets of Real numbers  Set A is called the domain of the function  The range of f is the set of all possible values of f(x) as x varies throughout the domain

5. Some Examples of Functions  Name a relationship in which the value of one quantity depends on another.

6. Morning Coffee  What do you think the graph of the temperature of your morning coffee would look like over time?

7. Storage Container  A rectangular storage container with an open top has a volume of 10m 3. The length of the base is twice its width. Material for the base costs $10 per square meter; material for the sides costs $6 per square meter. Express the cost of materials as a function of the width of the base.

8. A Tax Idea  In 2000, Presidential candidate Steve Forbes proposed a “flat tax” model: 0% on the first $36,000 and 17% on the rest.  Graph tax rate and tax owed versus income for incomes ranging from $0 to $80,000.  Can we come up with an equation for this situation?

9. 4 Representations of Functions 1. Verbally – by a description in words 2. Numerically – by a table of values 3. Visually – by a graph 4. Algebraically – by an explicit formula

10. Domain and Range  Find the domain and range of each function:

11. Vertical Line Test  A curve in the xy-plane is the graph of a function of x if and only if no vertical line intersects the curve more than once.

12. TI-89 Graphing Basics  To graph a function its equation must be entered. Press ◊ F1 to type the equation.  Pressing ◊ F3 displays the graphs of the equations defined and selected.  Press ◊ F2 to see the size of the graphing window.

13. Piecewise Functions

14. Absolute Value

15. Symmetry: Even and Odd Functions  Even Function: If a function f satisfies f(-x) = f(x) for every number x in its domain, then f is called an even function  Odd Function: If a function f satisfies f(-x) = -f(x) for ever number x in its domain, then f is called an odd function

16. Odd, Even, or Neither

17. Odd, Even or Neither

18. Group Work 1  How Cool is your Chicken?

19. Homework  Some problems to try for tomorrow (will not be collected)  Page 22 – 1, 5-8, 14, 18, 23, 25, 27, 37, 39, 47, 53, 61, 63, 65