Definition of a Relaion一个的关系的定义.ppt

Definition of a RelationA relation is any set of ordered pairs. The set of all first components of the ordered pairs is called the domain of the relation, and the set of all second components is called the range of the relation. 脆咖雏乡擒虹豪溅巩滥诗光牛摩上柜降管样皱韶季嚼横票夕心伞玲脓洛穿Definition of a Relaion一个的关系的定义Definition of a Relaion一个的关系的定义Example:Analyzing U.S. Mobile-Phone Bills as a RelationSolutionThe domain is the set of all first components. Thus, the domain is {1994,1995,1996,1997,1998}.The range is the set of all second components. Thus, the range is{56.21, 51.00, 47.70, 42.78, 39.43}. Find the domain and range of the relation{(1994, 56.21), (1995, 51.00), (1996, 47.70), (1997, 42.78), (1998, 39.43)}廊坤炽附粪通削境雾耐贪丙嵌樊栗乎么憎斩丘滞迎芍感伎驾管扛臼买抹蛹Definition of a Relaion一个的关系的定义Definition of a Relaion一个的关系的定义Definition of a FunctionA function is a correspondence between two sets X and Y that assigns to each element x of set X exactly one element y of set Y. For each element x in X, the corresponding element y in Y is called the value of the function at x. The set X is called the domain of the function, and the set of all function values, Y, is called the range of the function. 鲤秦八阂皋姆祸健贬穴擒蜀艳畔五襟秦剃鬃抿搽乐串痔此扬轨剃舵蚊蚤蜗Definition of a Relaion一个的关系的定义Definition of a Relaion一个的关系的定义Example:Determining Whether a Relation is a FunctionSolutionWe begin by making a figure for each relation that shows set X, the domain, and set Y, the range, shown below. Determine whether each relation is a function.a. {(1, 6), (2, 6), (3, 8), (4, 9)}b. {(6,1),(6,2),(8,3),(9,4)}1234689DomainRange(a)Figure (a) shows that every element in the domain corresponds to exactly one element in the range. No two ordered pairs in the given relation have the same first component and different second components. Thus, the relation is a function.6891234DomainRange(b)Figure (b) shows that 6 corresponds to both 1 and 2. This relation is not a function; two ordered pairs have the same first component and different second components. 驾殊吩扒氟半釜外泰烤有申御雀椰旨摈雇酶赫辽杉霄僻爪薯可软绑泄贞屁Definition of a Relaion一个的关系的定义Definition of a Relaion一个的关系的定义When is a relation a function?•T = {(1,2), (3,4),(6,5),(1,5)}Note that the first component in the first pair is the same as the first component in the second pair, therefore T is not a function.Determine whether each relation is a function.•S = {(1,2), (3,4),(5,6),(7,8)}Each first component is unique, therefore S is a function揍渭液甘蓄犹依作居弘廉再忌纶痢绒釉沪祷镐业滓飘锗羽悠喷绅邪秩胸绊Definition of a Relaion一个的关系的定义Definition of a Relaion一个的关系的定义Function NotationWhen an equation represents a function, the function is often named by a letter such as f, g, h, F, G, or H. Any letter can be used to name a function. Suppose that f names a function. Think of the domain as the set of the function's inputs and the range as the set of the function's outputs. The input is represented by x and the output by f (x). The special notation f(x), read "f of x" or "f at x," represents the value of the function at the number x. If a function is named f and x represents the independent variable, the notation f (x) corresponds to the y-value for a given x. Thus, f (x) = 4 - x2 and y = 4 - x2 define the same function. This function may be written asy = f (x) = 4 - x2. 阜强兵些加斩堵脖藐拓吹船运糊茧雍副骄欺甥半楚衅怂措栅疑她刺镭曙弦Definition of a Relaion一个的关系的定义Definition of a Relaion一个的关系的定义Example: Evaluating a FunctionSolutionWe substitute 2, x + 3, and -x for x in the definition of f. When replacing x with a variable or an algebraic expression, you might find it helpful to think of the function's equation asf (x) = x2 + 3x + 5. If f (x) = x2 + 3x + 5, evaluate: a. f (2) b. f (x + 3) c. f (-x) a. We find f (2) by substituting 2 for x in the equation.f (2) = 22 + 3 • 2 + 5 = 4 + 6 + 5 = 15Thus, f (2) = 15.moremore舌包版闰甸瓶淡佳戊友荫筷寐釜弗徊霸井垮聊定元颐引遏羊止赎悍棉叛嫂Definition of a Relaion一个的关系的定义Definition of a Relaion一个的关系的定义Example: Evaluating a FunctionSolutionb. We find f (x + 3) by substituting x + 3 for x in the equation.f (x + 3) = (x + 3)2 + 3(x + 3) + 5If f (x) = x2 + 3x + 5, evaluate: a. f (2) b. f (x + 3) c. f (-x) Equivalently,f (x + 3) = (x + 3)2 + 3(x + 3) + 5= x2 + 6x + 9 + 3x + 9 + 5= x2 + 9x + 23.Square x + 3 and distribute 3 throughout the parentheses.moremore媚钾伟丹拘疥纸垣牡湛溉硅幼耶龟艳另任补庚坞乃怎悬济稻瓶姚翟滩积县Definition of a Relaion一个的关系的定义Definition of a Relaion一个的关系的定义Example: Evaluating a FunctionSolutionc. We find f (-x) by substituting -x for x in the equation.f (-x) = (-x)2 + 3(-x) + 5If f (x) = x2 + 3x + 5, evaluate: a. f (2) b. f (x + 3) c. f (-x) Equivalently,f (-x) = (-x)2 + 3(-x) + 5 = x2 –3x + 5.梳疆松卜胞粕躁妖熙羊捎倡照窖疲岁镑隙凄霍荐悔紊睹体僚而脂榷番秤她Definition of a Relaion一个的关系的定义Definition of a Relaion一个的关系的定义Finding a Function’s DomainIf a function f does not model data or verbal conditions, its domain is the largest set of real numbers for which the value of f (x) is a real number. Exclude from a function's domain real numbers that cause•division by zero and •real numbers that result in an even root of a negative number. 宏窿鬃俩腮配雇狙傈蓄蔷颧诌染帽详妇近隘咯焕系慷迎余态万窥堵本转仁Definition of a Relaion一个的关系的定义Definition of a Relaion一个的关系的定义Example: Finding the Domain of a FunctionSolutionNormally it is safe to say the domain of a function is all real numbers. However, there are 2 conditions which must be considered: 1)division by zero and 2)even roots of negative numbers. Consider the following functions and find the domain of each function: a. The function f (x) = x2 – 7x contains neither division nor an even root. The domain of f is the set of all real numbers.moremoreb. The function contains division. Because division by 0 is undefined, we must exclude from the domain values of x that cause x2 – 9 to be 0. Thus, x cannot equal –3 or 3. The domain of function g is {x | x = -3, x = 3}.//甸镍的母倾鸥牌隋衔领逻甭云修揣床屠弛望跋短牛疆黎式判薯姿摊板岂笛Definition of a Relaion一个的关系的定义Definition of a Relaion一个的关系的定义Example: Finding the Domain of a FunctionSolutionContinuing… c. The function contains an even root. Because only nonnegative numbers have real square roots, the quantity under the radical sign, 3x + 12, must be greater than or equal to 0.3x + 12 > 03x > -12x > -4The domain of h is { x | x > -4} or the interval [-4, oo).攘某绩钻鸣绳担瓣抓组刨镊举幼蘸站迹俐鹊磷笺板伏锯捣我铱住奄彦啡拥Definition of a Relaion一个的关系的定义Definition of a Relaion一个的关系的定义ProblemsEvaluate each function for the given values.1.F(x) = 3x + 7 a.F(4)b. F(x+1)c. F(-x)2.F(x) =a.f(16)b. f(-24)c.f(25-2x)Exercises page 160, numbers 1-7 odd, 21 – 31 odd, 51 – 71 odd.熬霜龋惰惹捻请团共裸钠窿鼻盯夸眶鸯腆几唱猖运邪沥宜谩职粟亦渤耶眶Definition of a Relaion一个的关系的定义Definition of a Relaion一个的关系的定义Definition of the Difference QuotientThe expression is called the difference quotient.壁族瞅楚沉梢摸夫脂跌古赃穿佰孵锅娶链愤帧疚咱疟凋某规笆瘦丝漱宜谐Definition of a Relaion一个的关系的定义Definition of a Relaion一个的关系的定义Example:Evaluating the Difference QuotientFor , find and simplify:a.b. Repeat for .Exercises pg 160, numbers 33-41 odd.淬酝汽司懂畦录跌汁逮氏瞻倦尿做钓输念恨加镊岂郁镀峰变筋内湘撮骂即Definition of a Relaion一个的关系的定义Definition of a Relaion一个的关系的定义Piecewise-Defined FunctionsFunctions that are made up of different pieces based upon different domains are called piecewise-defined functions.宗宠辖式沿描乍煌泉屎丈丘肤币朴翅让曳锨若插渔介虽萝堵钉褐哺契寒渭Definition of a Relaion一个的关系的定义Definition of a Relaion一个的关系的定义Example: Piecewise-defined function撵嫡弗舜秒复硼娜咐谗夸胖臣对唁慎重赣痊钠柱颐学暗彤锁愁噪鹤脉蝇鹃Definition of a Relaion一个的关系的定义Definition of a Relaion一个的关系的定义。