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1 . Graphs of Functions 69. GRAPHS OF FUNCTIONS One can envisage that mathematical theor will go on being elaborated and etended indefinitel. How strange that the results of just the first few centuries of mathematical endeavor are enough to achieve such enormousl impressive results in phsics. P.W.C. Davies Graph of a Function A function f assigns a range element to each domain element, so it is often useful to think of the function as pairing numbers (if the domain and range are sets of numbers). The function is completel determined b these ordered pairs. In mathematical notation, f, f D, where D is the domain of f. When the function is defined b an equation, f, then f, f, D. The ordered pairs that define f look like coordinates of points in the plane, and we can use this feature to define the graph of a function. Definition: graph of a function If f is a function with domain D, thenthegraphoffis the set of points with coordinates, such that D and f. An accurate graph of a function makes both the domain and the range of the function apparent. The domain is the set of values of points on the graph, and the range is the set of values. Function Properties and Graphs I think that starting mathematics earl had given me a certain selfreliance. I felt ou didn t learn anthing in class, ou just figured it out ourself. Paul Cohen It is natural to think of drawing a graph b plotting points and connecting them in an appropriate wa to get a curve. This is precisel how a computer or graphing calculator shows graphs on a screen. Without the capabilit to compute hundreds of function valuesquickl, however, pencil and paper techniques are time consuming and tedious. With or without access to the tools of technolog, some additional tools can help us draw graphs and understand the properties of functions. In this section, we eamine smmetr properties of graphs and introduce the notion of even and odd functions. Also, certain core graphs are given. Knowing a single core graph and how it is affected b simple changes, we can draw graphs of a whole famil of related functions. Core Graphs There are a few graphs that should be familiar to ever precalculus student. We will do lots of graphing with calculators and the aid of technolog, but ou should be able to sketch each of the following core graphs without an help. Knowing the properties of these simple functions and some of their ke features will make discussions of all kinds of functional behavior more meaningful. Use our graphing calculator as needed to help absorb the ideas, but make ourself confident that ou know the graphs of the functions in Figure.
2 70 Chapter Functions (a) f()= (b) f()= (c) f() = (d) f()= (e) f()= (f) f()= FIGURE Catalog of core graphs Intercepts, Smmetr, Even and Odd Functions We are alwas interested in the points where a graph meets the coordinate aes. If 0 is in the domain of f, thenf 0 is called the intercept and the point 0, f 0 is the intercept point. A graph need not meet the ais, but if it does, an points where it does are called intercept points, and if f c 0, then the number c is called an intercept. Given a point A a, b, three smmetricall located points can help in graphing. We can reflect A in the ais to the point C a, b, intheais to the point D a, b or in the origin to the point E a, b. Figure a shows a firstquadrant point A and its reflections. The graph of a function ma have all sorts of smmetr properties (or none). Suppose that for ever arbitrar point a, b on the graph of a function, the graph also contains the reflection of a, b in the ais. That is, whenever a, b is on the graph, a, b isalsoonthegraph.thenwesathatthegraphissmmetric about the ais and that the function is an even function. In functional notation, since to sa that a, b is on the graph means that f a b, a function is even when f f. See Figure b. If, whenever a, b is on the graph of f, a, b is also on the graph, then the graph is smmetric about the origin and the function is an odd function. In functional notation, a function is odd if f f for ever in D. See Figure c.
3 . Graphs of Functions 7 Reflection in ais A(a, b) C( a, b) ( a, b) (a, b) (a, b) E( a, b) 0 Reflection in ais Reflection in origin D(a, b) ( a, b) (a) Point Reflections (b) Even; Smmetric about ais (c) Odd; Smmetric about origin FIGURE The catalog of core graphs in Figure has eamples that should help ou remember the distinction between even and odd functions. The parabola and the absolute value function are even; the graphs are clearl smmetric about the ais. The line, the cubic, and the reciprocal are all smmetric about the origin; the functions are odd. The square root function is neither odd nor even. We sum up in the following. Definition: even and odd functions, smmetr properties Suppose f is a function with domain D. If, for ever in D, wehave f f,thenfis an even function; f f,thenfis an odd function. The graph of an even function is smmetric about the ais. The graph of an odd function is smmetric about the origin. In addition to just looking at a single graph, graphing calculators can be used to see whether or not f f or f f ; compare the graph of f with the graph of f ( ), where we replace each in f b. EXAMPLE Even and odd functions Sketch the graph and determine, both graphicall and algebraicall, whether the function is odd or even. (a) f ( (b) g( (a) Algebraic f, so f f, and b definition, f is an even function. Graphical The graph, shown in Figure (a), is smmetric about the ais and is clearl the graph of an even function. (b) Algebraic For the function g, wehaveg g, and so from the definition, g is an odd function. Graphical The calculator confirms that the graph is smmetric about the origin, and hence the graph of an odd function. See Figure b.
4 7 Chapter Functions = + = (a) Even Function (b) Odd Function FIGURE A function ma be neither even nor odd, as the net eample shows. EXAMPLE Neither even nor odd Show that the function is neither even nor odd, and sketch its graph (a) F (b) G Appl the definitions for even and odd functions. F F G G Since F F and F F, F is neither even nor odd, and similarl, G is neither even nor odd. To graph the two functions, plot points as in Figure 5. (, 0) = 5 6 = (, ) (, ) (, ) (0, ) (a) F is neither even nor odd. (b) G is neither even nor odd. FIGURE 5
5 . Graphs of Functions 7 Piecewise, Step Functions, and Calculator Graphs We have alread seen some eamples of an important class of functions called piecewise functions because the are defined in pieces (with different rules for different portions of their domains). The first eample we encountered is the absolute value function, even though we did not recognize it when we first introduced absolute values. Among the properties of absolute values in Section. we listed the equalit, which can be justified b considering values, but it ma also be helpful to get calculator reinforcement. Graph both Y ABS(X) and Y X and see that the graphs are identical. The function can also be written in pieces: if 0 if 0 The graph consists of part of the line and part of the line. You ma want to refer again to the Technolog Tip in Section. to make sure that ou know how to graph piecewisedefined functions on our calculator. (, ) [, ] b [.5,.5] FIGURE 6 EXAMPLE Calculator graphs of piecewise functions Draw a calculator graph of the piecewise function from Eample of Section., f if if The graph is shown in Figure 6. You ma wish to eperiment with different windows to see how changing the view alters the shape of the pieces we see. Whatever the window, however, the graph of f consists of two pieces that meet at the point (, ). Greatest Integer Function Another piecewisedefined function that is useful in man different contets is the greatest integer function, which is programmed into most graphing calculators, defined as the largest integer that is less than or equal to. An real number is either an integer or it lies between two integers. If n n, the largest integerlessthanorequaltois n, soforsuchan,int n. The function is denoted b Int on graphing calculators, or sometimes Floor, or, in older books, b. Definition: The greatest integer function The greatest integer function of, denoted b Int, isthe largest integer that is less than or equal to. As eamples, Int, and for an integer n, Int n n, Int 0. 0, and for an number between 0 and, Int 0, Int because is the largest integer less than, Int because is between and. Postal rates are eamples of functions defined piecewise, where the definition can make use of Int. Mailing cost is a function of the weight of a letter. It remains constant for a while and then suddenl jumps to a new value.
6 7 Chapter Functions (, 78) (, 55) (0, ) (, 0) 5 FIGURE 7 C W Int W EXAMPLE The postage function In 995, postal rates for first class letters delivered within the United States were set as follows: the cost is cents for anthing less than ounce; foreach additional ounce (up to ounces) the cost increases in increments of cents. Epress the cost C (in cents) of firstclass postage as a function of the weight W (ounces) and draw a graph. In mathematical language, we can epress the cost C as a function of the weight W either piecewise, or b using the greatest integer function. C Int W, 0 W. The graph of the cost function is shown in Figure 7. TECHNOLOGY TIP Graphing in dot mode In connected mode, a calculator graph of a piecewisedefined function can make it appear as if there is a vertical piece of the graph connecting pieces that should not be connected. Don t be fooled. There can be a jump from one piece to another between piels. When values differ in adjacent piel columns, the calculator connects piels in vertical columns joining separated points. It is often helpful to change to a nonconnected format to see jumps in graphs of piecewise functions. In calculator graphs, remember that unless the calculator is in dot mode, the calculator connects separated piels in adjacent columns. Not all equations define functions, and some familiar graphs are not graphs of functions. A function assigns to each domain element eactl one element of the range. This implies that for an function f and an given domain number c, there is eactl one point, c, f c, onthegraphoff. A vertical line can meet the graph of f in at most one point, which is the basis for the following hand test. Vertical line test For a given graph, if at each number c of the domain, the vertical line c intersects the graph in eactl one point, then the graph represents the graph of a function. If some vertical line meets a graph in more than a single point, then the graph is not the graph of a function. [, ] b [, ] FIGURE 8 Vertical line meets graph in two places. EXAMPLE 5 Vertical line test Useacalculatortosketchagraphofall points that satisf the equation. Use the vertical line test to verif that the graph is not that of a function. In function mode, we cannot enter equations ecept in the form of Y...,sowhen we solve the given equation for,weget. On the same screen we graph Y X and Y X. Thetwographstogetherformaparabolaopeningtotheright, as shown in Figure 8. Since an vertical line through the positive ais meets the graph twice, the vertical line test tells us that the graph is not the graph of a function.
7 . Graphs of Functions 75 EXERCISES. Check Your Understanding Eercises 6 True or False. Give reasons.. The graph of a function cannot have more than one  intercept point.. The graph of isidenticaltothegraphof.. For the greatest integer function f Int, (a) f.5 f.5 (b) f f.. The distance between the  andintercept points of the graph of is. 5. For an function f, the function g f is an even function. 6. For an even function f, if(, ) is on the graph of f, then (, ) must also be on the graph of f. Eercises 7 0 Fill in the blank so that the following statement is true. If ou draw a graph of function f using [ 0, 0] [ 0, 0], then the number of intercept points shown in the displa is. 7. f f f 0. f 6 Develop Master Eercises Isolated Points A function is given along with its domain. Draw a graph of the function. The graph consists of isolated points. State the range of the function.. f ; D,,. f ; D, 0,,. f ; D,, 0,,. f ; D,, Eercises 5 8 Make a table of several, ordered pairs that satisf the equation. Plot the points in our table and drawagraph Eercises 9 Find the value of or so that point P is on the graph of f. 9. f ; P, 0. f ; P,. f ; P,. f 8; P, 5 Eercises 6 Odd, Even Determine whether function f is odd, even, or neither. Do the same for g. First draw a graph then use algebra to support our conclusion.. f, g. f, g 5. f, g 6. f, g Eercises 7 8 Graphs Draw a graph of f. Give the coordinates of the  and intercept points. 7. f 8. f 9. g 0. g. f. g 6. g. g 5. f 6. g 7. g 8. f 8 Eercises 9 0 Calculator Graph Suppose ou are interested in using a graph to help ou get information about the zeros of f. Which of the given windows would ou use? 9. f 7 75 (i) 0, 0 0, 0 (ii) 0, 0 00, 00 (iii) 0, 0 00, f 5 00 (i) 0, 0 0, 0 (ii) 6, 0 5, 0 (iii) 0, 0 5, 00 Eercises Window Dimensions The graph of f has two intercept points and either a highest or lowest point. Give the dimensions of a window for which the graph of f will show this information. (Answers ma var.). f f 0 8. f 5. f 8 0 Eercises 5 8 Limited Domain (a) Draw a graph of f, with the indicated domain. (b) Find the range of f. 5. D 0 ; f 6. D 0 ; f 7. D ; f 8. D ; f
8 76 Chapter Functions Eercises 9 Piecewise Graph Draw a graph of the given function and determine the range. 9. f,, 0. f. f if 0 if 0 if 0 if 0 if if 9. Use the vertical line test to determine which of these graphs are graphs of functions that have as the independent variable. Eercises 8 Refer to the function f whose graph is shown in the diagram with domain, 6. = f () 5 (a) (b). From the graph, give f, f 0, andf.. Order the following numbers from smallest to largest: f, f, f, f 9.. (a) What is the maimum value (the largest value) of f? (b) What is the minimum value (the smallest value) of f? (c) What is the range of f? 5. Give the coordinates of the highest and the lowest point on the graph. 6. Give the coordinates of the intercept point and the intercept points. 7. (a) Forwhatvaluesofis f negative? (b) Forwhatvaluesofis f positive? 8. True or false. Eplain. (a) f is less than f. (b) f. isanegativenumber. (c) f f isanegativenumber. (d) There are three intercept points for the graph of f. (c) Eercises 50 5 Interesting Functions (a) Give the domain for f and for g. Evaluate f at 0,,0, 0, 0, 56. (b) Draw graphs of f and g on separate screens. (c) Look at the graphs and describe an interesting features. (d) Are functions f and g identical? Eplain. (e) What is the solution set for f? f? 50. f 6 6 g 8 5. f g Eercises 5 5 Functions Involving Abs (a) Draw a graph of f. Use a decimal window. (b) Use the graph to find the solution set for f f 5. f Eercises 5 55 (a) Evaluate f at 5,, 0,.5. (b) Give a formula for f in piecewise form. (c) Usepart(b) to draw a graph of f. 5. f min, f ma, 7
9 . Graphs of Functions 77 Eercises Graphs Involving Int Draw a graph of f. Use dot mode and a decimal window. (a) What is the rangeoff?(b) What is the solution set for f 5? 56. f Int 57. f Int 58. f Int f Int Int Eercises 60 6 Equation Involving [] Draw a graph of f. Use dot mode and a decimal window. Find the solution set for the given equation. Note: [] is the same as Int (). 60. f ; f 6. f ; f 6. f ; f 6. f ; f 0 Eercises 6 67 Set Find the solution set for the open sentence. 6. Int 5Int 0. (Hint: Factor.) Int p where p is a prime. 67. Int Postage Costs This section discussed postage charges. When a parcel or letter eceeds ounces, a different rule applies for determining mailing cost as a function of weight. The rule depends upon mailing zones as well as weight and is given in tabular form. For eample, when mailing from Zone to Zone 8, the table below lists charges where w is the weight (not eceeding the number of pounds) and c is the cost in dollars. (a) Draw a graph of c as a function of w. (b) What is the cost of mailing a package that weighs pounds 5 ounces? pounds ounces? (c) If the cost of mailing a package is $6.00, what do we know about its weight? w c Pounds Dollars Bug on a Ladder A bug starts at point (, 0) and travels along the line segment AB toward point (0, ) as shown in the diagram. If P, denotes the location of the bug when it has traveled a distance d from (, 0), epress the coordinates and as functions of the distance d. Eercises 70 7 B(0, ) d Parking Costs Bug has crawled a distance d to point P(, ) A(, 0) 70. A parking garage charges $.00 for parking up to one hour and $0.50 for each additional hour (or fraction thereof ), with a maimum of $8.00 if ou park hours or longer. Suppose denotes the number of hours ou park and (dollars) the corresponding cost. Then is a function of given in piecewise form: 0.50 Int 8 if if (a) Draw a graph of this function. Use dot mode. (b) If ou have onl $5.00, how long can ou park? 7. A parking garage charges $.00 for parking up to one hour and $0.0 for each additional fifteen minutes (or fraction thereof ), with a maimum of $0. Suppose denotes the number of hours ou park and (dollars) the corresponding cost. Then is a function of given in piecewise form: Int 0 if if 6.75 if 6.75 (a) Useseveralvaluesofto check this formula. (b) What is the cost for hours and 0 minutes? (c) Forwhatvaluesofis the cost $.50? Check using thegraph.usedotmodeandadecimalwindow.
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