Functions and Graphs: Definitions and Examples

1. Lesson 1.2: Functions and Graphs AP Calculus Mrs. Mongold

2. 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

3. Example 1: • Area of a circle is dependent on the radius, so A = A(r) • A(r) = r2 • 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) = (22) = 4

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 = x2 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 = x2, x>0

5. 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

6. Infinite intervals a a a a

7. Finite Intervals a b a b a b a b

8. Examples

9. Power Function for Graphing Activity • 1. y = mx values m = -1/3, -1, -2, 3, 1, ½ • 2. y = x2 • 3. y = x3 • 4. y = 1/x • 5. y = √x • 6. y = x1/3 • 7. y = 1/x2 • 8. y = x3/2 • 9. y = x2/3 • Graph each with your calculator, play with the window and determine the domain and range and any calculator errors you may experience!

10. 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 (-x2) = x2 • If y is an odd power of x, it is an odd function because (-x)3 = -x3

11. 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

12. 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 1800 about the origin leaves the graph unchanged

13. Example: Recognizing Even and Odd Functions • 1. f(x) = x2 • 2. f(x) = x2 + 1 • 3. f(x) = x • 4. f(x) = x+1