2ObjectivesStudents will be able to identify the domain and range of a function using its graph or equation.Students will be able to recognize even functions and odd functions using equations and graphs.Students will be able to interpret and find formulas for piecewise defined functions.Students will be able to write and evaluate compositions of two functions.
3Key Ideas Functions Domains and Ranges Viewing and Interpreting Graphs Even Functions and Odd functions – symmetryFunctions defined in piecesThe Absolute Value FunctionComposite Functions
4Functions Independent Variable vs. Dependent Variable Domain and Range Natural DomainBoundaries, boundary points, and interval notationFunctionsFunction notation
5Independent Variables vs. Dependent Variables The independent variable is the first coordinate in the ordered pair (the x values)The dependent variable is the second coordinate in the ordered pair (the y values)
6Domain The set of all independent variables (x-coordinates) If the domain of a function is not stated explicitly, then assume it to be the largest set of real x-values for which the equation gives real y-values.Any exclusions must be specifically stated.Natural DomainThe set of all non-restricted x-values
7Open vs. Closed Intervals The domains and ranges of many real-valued functions are intervals or combinations of intervals.These intervals may be open, closed, or half-open.
8There are 4 ways to express domains: Graph it:Name it:Use Set Notation:Use Interval Notation:
9What is set notation?Set notation is what you have used in the past. . .For example. . .x > 10-3 <x <23
10What is interval notation? Interval notation uses ( ,[, ), or ] to denote the set of numbers to which you refer.( or ) indicate open boundaries[ or ] indicate closed boundariesFor example: x > 10 would be (10,∞)-3 <x <23 would be [-3, 23]
11How does set notation compare to interval notation? Both are used to indicate sets of numbers
12Example: Are there any restrictions on x ? Because there is no restriction on the possible values that may be used for x, the natural domain is the set of all real numbers.How do you express this domain?
13Is the domain of our example An open or closed interval?Open intervals contain no boundary points.Closed intervals contain their boundary points.
14The 4 ways to express our domain: Graph it:Name it:The set of all real numbers.-∞ < x <∞Use Set Notation:Use Interval Notation:(-∞, ∞)
15How do you express this domain? Example:Are there any restrictions on x ?You cannot have a negative radicand.Therefore, natural domain is the set of all x values for which 2x – 8 0.How do you express this domain?
16The 4 ways to express our domain: Graph it:4Name it:The set of all real numbers greater than or equal to 4.4 < x <∞Use Set Notation:Use Interval Notation:[4, ∞)
17Range Range The set of all dependent variables (the y-coordinates) for which the function is defined
18What makes a relation a function? FunctionsWhat makes a relation a function?Consider functionsgeometrically&analytically
19Geometrically speaking. . . The graph must pass the vertical line test:Are the following functions?Can you explain why?
20Analytically. . . By Definition: A function from a set D to a set R is a rule that assigns a unique element in R to each element in D.DR