Lesson 1.2: Functions and Graphs AP Calculus Mrs. Mongold

Definitions Dependent variables: determined by the values of the variables on which they depend  Ex. Boiling temperature depends on elevation and interest earned depends on the interest rate Independent variables: variables that are depended on  Elevation when boiling water and interest rate when earning interest Domain: inputs Range: outputs Function: a rule that assigns a unique element in a set R to each element in a set D.  Ex. b = f(e) boiling point is a function is a function of elevation  I = f(r) Interest earned is a function of interest rate

Example 1: Area of a circle is dependent on the radius, so A = A(r)  A(r) =  r 2 Domain – the set of all possible radii ( positive real numbers) Range-all positive real #’s The value of the function at r = 2 is A(2) =  (2 2 ) = 4 

Domains and Ranges A function defined as y = f(x) and the domain is not stated expicitly or restricted by context, then the domain is assumed to be the largest set of x- values for which the formula gives real y-values  This is the so called natural domain y = x 2 domain is understood to be entire set of real numbers If we want to restrict values of x to positvie values we must write y = x 2, x>0

Types of Intervals The endpoints make up the interval’s boundary and are called boundary points The remaining points make up intervals interior and are called interior points Open intervals contain no boundary points Every point of an open interval is an interior point We use ( ) and [ ] for interval notation ( ) are used for an open interval, when we don’t or can’t include the endpoint [ ] are used for a closed interval, when we want to include the endpoint

Infinite intervals Name: The set of all real #’s Notation: -∞<x<∞ or (-∞, ∞ ) Name: all real numbers greater than a Notation: x> a or (a, ∞) Name: all real numbers great than or equal to a Notation: x> a or [a, ∞) Name: all real numbers less than a Notation: x<a or ( -∞, a) Name: all real numbers less than or equal to a Notation: x < a or ( -∞, a] a a a a

Finite Intervals Name: all real numbers greater than a less than b Notation: a<x<b or (a, b) Name: all real numbers greater than or equal to a and less than or equal to b Notation: a<x<b or [a, b] Name: all real numbers greater than or equal to a and less than b Notation: a<x<b or [a, b) Name: all real numbers greater than a and less than or equal to b Notation: a<x<b or (a, b] a b ab ab ab

Examples FunctionDomain (x)Range (y) y = x 2 y = 1/x y = √x y = √(4 – x) y = √(1-x 2 )

Power Function for Graphing Activity 1. y = mx values m = -1/3, -1, -2, 3, 1, ½ 2. y = x 2 3. y = x 3 4. y = 1/x 5. y = √x 6. y = x 1/3 7. y = 1/x 2 8. y = x 3/2 9. y = x 2/3  Graph each with your calculator, play with the window and determine the domain and range and any calculator errors you may experience!

Even & Odd Functions - Symmetry A function y = f(x) is an  Even function of x if f(-x) = f(x)  Odd function of x if f(-x) = -f(x) For every x in the functions domain. The names even and odd come from powers of x. If y is an even power of x it is an even function because (-x 2 ) = x 2 If y is an odd power of x, it is an odd function because (-x) 3 = -x 3

Graphs of Even and Odd Functions The graph of an even function is symmetric about the y-axis. Since f(-x) = f(x), a point (x, y) lies on the graph if and only if the point (-x, y) lies on the graph  Think parabola

Graphs of Even and Odd Functions The graph of an odd function is symmetric about the origin. Since f(-x) = -f(x), a point (x, y) lies on the graph if an donly if the point (-x, -y) lies on the graph  A rotation of about the origin leaves the graph unchanged

Example: Recognizing Even and Odd Functions 1. f(x) = x 2 2. f(x) = x f(x) = x 4. f(x) = x+1