1.2 Functions and Their Properties Name: Objectives: Students will be able to represent functions numerically, algebraically, and graphically, determine the domain and range for functions, and analyze function characteristics such as symmetry, extreme values and end behavior. A from a set D to a set R is a rule that assigns to every element in D a unique element in R. *Note: cannot repeat. The set D of all input values (x-values) is the of the function. The set R of all output values (y values) is the of the function. Examples Determine if the following are functions. 1.) A = {(0,1), (-9,10), (3,10)} 2.) A = {(0,1), (0,10), (3,10)} Jul 31 3:30 PM Note: f(x) is fancy for. Determine whether the formula determines y as a function of x. If not, explain. 1.) y = x 2 2.) x = y 2 3.) y = x 2 ± 4 4.) y = 2x - 1 Is there a way to look at a graph and determine if it's a function? Vertical Line Test: A graph [set of points (x,y)] in the xy-plane defines y as a function of x if and only if no vertical line intersects the graph in more than one point. Jul 31 3:38 PM 1
Examples Use the vertical line test to determine whether the curve is a graph of a function. 1.) 2.) 3.) 4.) Jul 31 3:44 PM Domain -The domain of a polynomial is always or. Examples of polynomials: -Restrictions to the domain come when we have or. Examples: Examples: Find the domain of the function algebraically and support your answer graphically. 1.) f(x) = x 3 + 1 2.) g(x) = 5 Jul 31 3:51 PM 2
3.) h(x) = x + 1 4.) m(x) = x (x + 1)(x - 1) 5.) p(x) = x - 2 6.) j(x) = 5 - x Jul 31 3:59 PM Range Examples: Find the range of the functions. 1.) f(x) = x 2 + 4 2.) g(x) = 5 - x 3.) h(x) = x + 2 + 3 4.) m(x) = x 2 4 - x 2 Jul 31 4:04 PM 3
A function is at a point if the graph does not break at that point. Note: You can draw a continuous function without lifting your writing utensil. Removable discontinuity: (Holes) The discontinuity can be patched by redefining the output value so as to plug the hole. Nonremovable discontinuity: (Jump discontinuity and asymptotes) The discontinuity is impossible to patch by redefining the output value. Jul 31 4:18 PM Jul 31 4:28 PM 4
Examples Graph the functions determine any points of discontinuity. If there are any discontinuities, tell whether they are removable or nonremovable. Note: It might be helpful to look at the algebraic representations, as well. 1.) f(x) = 5 2.) g(x) = x 2 + x x x 3.) g(x) = -1, x > 0 4.) k(x) = 3 1, x 0 x-1 Jul 31 4:25 PM Increasing, Decreasing and Constant Functions Examples: Determine the intervals where the functions are increasing, decreasing or constant. 1.) 2.) g(x) = x + 1 + x - 1-3 Jul 31 4:51 PM 5
Boundedness Jul 31 4:57 PM Examples Determine whether the function is bounded above, bounded below, or bounded on its domain. 1.) y = 19 2.) y = x 2 + 5 3.) y = (1/2) x 4.) y = x 5 + x - 1 5.) y = - 4-x 2 Jul 31 8:41 PM 6
Extrema A of a function f is a value f(c) that is greater than or equal to all range values of f on some open interval containing c. If f(c) is greater than or equal to all range values of f, then f(c) is the (or ) value. A of a function f is a value f(c) that is less than or equal to all range values of f on some open interval containing c. If f(c) is less than or equal to all range values of f, then f(c) is the or ( ) value of f. Local extrema are also called. Jul 31 8:45 PM Examples Identify the extrema. 1.) 2.) 3.) g(x) = x 3-4x + 1 Jul 31 9:06 PM 7
Symmetry Symmetry with respect to (w.r.t) the y-axis Example: f(x) = x 2 Numerically Graphically Algebraically x f(x) For all x in the -3-2 -1 0 1 2 3 Aug 1 11:26 AM Symmetry w.r.t the x-axis Example: x = y 2 Numerically Graphically Algebraically x y Graphs with this 9 4 1 0 1 4 9 Aug 1 11:35 AM 8
Symmetry w.r.t the origin Example: f(x) = x 3 Numerically Graphically Algebraically x f(x) For all x in the -3-2 -1 0 1 2 3 Aug 1 11:41 AM Examples State whether the function is odd, even or neither. Support graphically and confirm algebraically. 1.) h(x) = 1/x 2.) f(x) = 3 3.) f(x) = x 3 + 0.04x 2 + 3 Aug 1 11:52 AM 9
Asymptotes -Vertical asymptotes: A vertical line ( ) that the function approaches, but never reaches. To find them, simplify the function completely. Set the denominator = 0 and solve. -Horizontal asymptotes: A horizontal line ( ) that the function approaches, but never reaches. Cases: 1.) If the top power is bigger, there are no H.A. 2.) If the top and bottom powers are the same, the H.A. is the ratio of the leading coefficients. 3.) If the bottom power is bigger, the H.A. is y = 0. Aug 1 11:54 AM Examples Find all of the horizontal and vertical asymptotes. 1.) f(x) = 6x 2.) g(x) = 5x + 2 2x + 1 x 3 + 27 3.) h(x) = 4x - 20 4.) k(x) = 6 x x 2-25 5.) j(x) = (1.2) x - 3 Aug 1 12:01 PM 10
End Behavior Limit Notation: lim f(x) = means as x goes to infinity, where x--> does f(x) go? lim f(x) = means as x goes to negative x--> - infinity, where does f(x) go? Examples: Determine the end behavior. 1.) 2.) 3.) f(x) = 4 Homework: pages 102-105: 1-8, 9-15 odd, 17-24, 25, 28, 29-33 odd, 35-40, 41-53 odd, 55, 58, 61, 62, 63-66, 67, 71-72 Aug 1 1:46 PM 11