1.1 Functions and Change Friday, August 26, 2005
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
What is a Function? Arise whenever one quantity depends on another
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
Some Examples of Functions Name a relationship in which the value of one quantity depends on another.
Morning Coffee What do you think the graph of the temperature of your morning coffee would look like over time?
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.
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?
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
Domain and Range Find the domain and range of each function:
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.
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.
Piecewise Functions
Absolute Value
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
Odd, Even, or Neither
Odd, Even or Neither
Group Work 1 How Cool is your Chicken?
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