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1.3 Representations of Functions 31 SpeciÞc Example General Statement Symbolic Representation is a square is a square If is a rectangle with sides of equal length is a square If is a square if and only if if equivalently then only if equivalently then is a rectangle with sides of equal length is a rectangle with sides of equal length is a square is a rectangle with sides of equal length is a rectangle with sides of equal length if and only if if equivalently if then only if equivalently if then The graphs of some functions are given in the Þgure below. f(V) A height V (r, A(r)) A(r) C(V) V r volume (a) f(V) = (b) A(r)=2 Figure 1.16 (c) the calibration function C(V) for the evaporating flask Graphsofmappings,,and fromExercise1.1aregiveninFigure1.17.Youshould now be able to check visually that and are functions while is not a function. Q R S (3, Q(3)) = (3, 6) 16 15 14 13 12 11 1 3 6 (3, R(3)) = (3, ) x 1 3 6 4 2 —2 1 3 6 x —4 6 x —6 (a) (b) (c) Figure 1.17 32 CHAPTER 1 Functions Are Lurking Everywhere EXERCISE 1.3 TwoofthefourgraphsgiveninFigure18arethegraphsoffunctions.15 Whichtwoarethey? Can you come up with a rule for determining whether or not a given graph is the graph of a function? y y y y (a) (b) (c) (d) x x x x Figure 1.18 Answer We can tell that the relationships represented in Figures 1.16(a)—(c), 1.17(a) and (b), and 1.18(b) and (d) are, in fact, functions. The test for a function is that every input must have only one output assigned to it; graphically, this means if we draw a vertical line through any point on the horizontal axis, this line cannot cross the graph in more than one place. (The vertical line will not cross the graph at all if it is drawn through a point on the horizontal axis that is not a valid input, i.e., not in the domain of the function.) This test for deciding if a graph represents a function is called the vertical line test. y y y y (a) (b) (c) (d) x x x x Not the graph of a function Not the graph of a function Figure 1.19 Vertical line test LetÕsreturntotheproblemofcalibratingbottles.Whenwecalibrateabottleweputina known volume of liquid and the calibration function (produces a height. Once we have calibrated a bottle we can then use it as a measuring device. We turn the procedure around so that the height becomes the input while the volume is the output. The key ingredient that makes the original volume-height assignment useful is that not only does each volume correspond to one height but each height corresponds to only one volume. The calibration function is 1-to-1. A function that is 1-to-1 has an inverse function that ÒundoesÓ . The domain of the inverse function is the range of ; if maps a to b then its inverse function maps b to a.16 EXERCISE 1.4 Given the graph of a function, how can we determine whether the function is 1-to-1? 15 The conventions about hollow and Þlled circles are the same as they are for interval notation. A hollow circle denotes a point that is not on the graph, while a Þlled circle indicates that the point is on the graph. 16 Sometimes people mistakenly think of the words ÒinverseÓ and ÒreciprocalÓ as being the same. The inverse of the function undoes . If adds 3 to its input then its inverse function subtracts 3; if multiplies its input by 3 then its inverse function divides by 3. If maps 7 to 11 then its inverse function maps 11 to 7. On the other hand, the reciprocal of is simply 1+. 1.3 Representations of Functions 33 Answer Bythehorizontallinetest:Ifanyhorizontallineintersectsthegraphinmorethanoneplace, then the function is not 1-to-1; if no horizontal line intersects the graph in more than one place, then the function is 1-to-1. Functions: The Grand Scheme In this text we will be looking at functions of one variableÑbut not all functions are functions of one variable. For instance, suppose you were to calculate the value, !, of the change in your pocket. ! is a function of ,, , , and , where ,, , , and are the number of quarters, dimes, nickels, and pennies, respectively, in your pocket. We can write this function as ! ,, , , 25, 10 5. As another example, suppose we denote by ! the volume of a Þxed mass of gas, by its temperature on the Kelvin scale, and by its pressure. Then the combined gas law tells us that ! #, where # is a constant, or, equivalently, ! # . We see that the volume of the gas will depend upon both the pressure and the temperature. ! is a function of two variables, ! , . If pressure is held constant, then volume is directly proportional to temperature; as the temperature of a gas goes up, the volume goes up as well. ! (a constant) is the gas law of Charles and Gay-Lussac. In other words, if pressure is held constant, then ! can be expressed as a function of one variable. Think back to Example 1.9 where we were expressing the volume of water in a conical coffeeÞlterasafunctionoftheheightofthewater.Webeganbywriting! 1"2, where both ", the radius of the liquid, and , the height of the liquid are variables. We wanted to express! asafunctionofonly.InthisexamplewecannotsimplysayÒlet" beaconstant,Ó since cannot vary without " varying. Instead, we found a relationship between and " that allowed us to express the former in terms of the latter, resulting in a function of only one variable. Because our focus will be on functions of one variable, when we do modeling we need to determine one independent variable and one dependent variable, for a total of two variables. If we appear to have more variables, then it is necessary to do one of two things: Figure out how we can express one of the variables in terms of another, or be less ambitious with our model, treating some quantities as constants when it is reasonable to do so. 34 CHAPTER 1 Functions Are Lurking Everywhere P R O B L E M S F O R S E C T I O N 1 . 3 1. (a) Considerthefollowinggraphs.Foreachgraphdecidewhetherornot isafunction of . If it is a function, determine the range and domain. y y y (a) 5 (b) 5 (c) 5 4 3 2 1 -5 -4 -3 -2 -1 1 2 3 4 5 6 x -1 -2 4 3 2 1 -3 -2 -1 1 2 3 4 5 6 x -1 -2 -3 4 3 2 1 -4 -3 -2 -1 1 2 3 4 5 6 7 x -1 -2 -3 -4 y y y (d) 3 (e) 3 (f) 3 2 2 2 1 1 1 -5 -4 -3 -2 -1 1 2 3 4 5 6 x -1 -2 -5 -4 -3 -2 -1 1 2 3 4 5 6 x -1 -2 -5 -4 -3 -2 -1 1 2 3 4 5 6 x -1 -2 y y y (g) (h) (i) 5 5 5 4 4 4 3 3 3 2 2 2 1 1 1 -5 -4 -3 -2 -1 1 2 3 4 5 6 x -1 -2 -5 -4 -3 -2 -1 1 2 3 4 5 6 x -1 -2 -5 -4 -3 -2 -1 1 2 3 4 5 6 x -1 -2 (b) Are any of the functions from part (a) 1-to-1? If so, which ones? In Problems 2 through 4 Þnd the domain of each of the functions. 2. (a) 2 (b) 24 1.3 Representations of Functions 35 3. (a) 4. (a) 1 (b) 3 (c) 2 4 (b) 1 (Hint: Keep in mind that if the numerator and denominator of a fraction are both negative, then the fraction is positive.) 5. Find the range of the functions in Problems 2 through 4. 6. Let be the function graphed below with domain [5, 4]. Use the graph to answer the following questions. (a) What is the range of ? (b) Is 1-to-1? (c) Where does take on its highest value? Its lowest value? (d) What is the highest value of ? (e) What is the lowest value of ? (f) For what is 0? (g) Approximate 0. (h) Approximate 4. (i) Approximate and . (j) For approximately what values of is 1? (k) For approximately what values of is %1? g (—3, 3) 3 2 1 —5 —4 —3 —2 —1 —1 (—5, —2) —2 x 1 2 3 4 (4, —1) 7. Two functions are equivalent if they have the same domain and the same input/output relationship. The Þrst function listed on each line below is called . Which of the functions listed on each line are equivalent to ? The domain of each function is the set of all real numbers. (Be careful to think about the sign of each function.) (a) 22 (b) 2 12 (c) 2 22 1 2 2 .// -22 1 22 2 2222 1 22 0 ... - --nqh--