2.3 Properties of Functions 2011 September 14, 2011 2.3 Properties of Functions Objectives: • Determine even and odd functions. • Locate and identify local maxima and minima. Oct 10­10:57 AM 1 2.3 Properties of Functions 2011 September 14, 2011 REVIEW: Find the domain of each function. Linear Function: Quadratic Function: Cubic Function: Sep 27 ­ 5:48 PM 2 2.3 Properties of Functions 2011 September 14, 2011 REVIEW: Find the domain of each function. Radical Function: Rational Functions: 1. 2. Sep 27 ­ 5:48 PM 3 2.3 Properties of Functions 2011 September 14, 2011 A. Determine Even and Odd Functions 1. A function is even if, for every (x, y) on the graph of f(x), there is also a (­x, y). (­2, 4) (2, 4) Oct 10­10:58 AM 4 2.3 Properties of Functions 2011 September 14, 2011 2. Testing to see if a function is even g(x) = x2 ­ 5 a. Graphically: Use a graphing calculator to see if it has y­axis symmetry. b. Algebraically: f(­x) = f(x) g(x) = x2 ­ 5 g(­x) = (­x)2 ­ 5 = x2 ­ 5 = g(x) c. Numerically: Test a value and its opposite. Let x = 2 g(2) = (2)2 ­ 5 = ­1 Let x = ­2 g(­2) = (­2)2 ­ 5 = ­1 Oct 10­3:38 PM 5 2.3 Properties of Functions 2011 September 14, 2011 3. A function is odd if, for every (x, y) on the graph of f(x), there is also a (­x, ­y). (2, 8) (1, 4) 4 2 ­2 ­2 ­4 1 2 (­1, ­ 4) (­2, ­8) Oct 10­11:01 AM 6 2.3 Properties of Functions 2011 September 14, 2011 4. Testing to see if a function is odd h(x) = 5x3 ­ x a. Graphically: Use a graphing calculator to see if it has symmetry with respect to the origin. b. Algebraically: f(­x) = ­f(x) h(x) = 5x3 ­ x h(­x) = 5(­x)3 ­ (­x) = ­5x3 + x = ­(5x3 ­ x) c. Numerically: Test a value and its opposite. Let x = 2 Let x = ­2 h(2) = 5(2)3 ­ 2 h(­2) = 5(­2)3 ­ (­2) = 5(8) ­ 2 = 5(­8) + 2 = 40 ­ 2 = ­40 + 2 = 38 = ­38 Oct 10­3:46 PM 7 2.3 Properties of Functions 2011 September 14, 2011 B. Increasing, Decreasing, and Constant Oct 10­11:02 AM 8 2.3 Properties of Functions 2011 September 14, 2011 Interval Notation ­5 ­4 ­3 ­2 ­1 ­5 < x < 2 ­5 ­4 ­3 ­2 ­1 ­5 < x < 2 0 1 is 0 2 3 4 5 [­5, 2] 1 is 2 3 4 5 (­5, 2) Oct 4 ­ 8:29 AM 9 2.3 Properties of Functions 2011 September 14, 2011 Interval Notation ­5 ­4 ­3 ­2 ­1 ­5 < x < 2 ­5 ­4 ­3 ­2 ­1 ­5 < x < 2 0 1 is 0 2 3 4 5 4 5 [­5, 2) 1 is 2 3 (­5, 2] Oct 4 ­ 8:32 AM 10 2.3 Properties of Functions 2011 September 14, 2011 1. Local Maxima: the y­value at a high point 2. Local Minima: the y­value at a low point 3. Example: (1, 2) (­1, 1) (3, 0) a. At what number(s) does f have a local maximum? b. What is the local maximum? c. At what number(s) does f have a local minimum? d. What are the local minima? e. List the intervals on which f is increasing. f. List the intervals on which f is decreasing. Oct 10­11:07 AM 11 2.3 Properties of Functions 2011 September 14, 2011 Homework: page 89 (11, 13, 15, 17, 19, 22, 24, 29, 31, 38, 40, 42, 63) Oct 10­3:57 PM 12