2.5 Library of Functions; Piecewise-defined Functions

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1 SECTION.5 Librar of Functions; Piecewise-defined Functions 07.5 Librar of Functions; Piecewise-defined Functions PREPARING FOR THIS SECTION Before getting started, review the following: Intercepts (Section., pp. 5 7) Graphs of Ke Equations (Section.: Eample, p. ; Eample, p. 9; Eample, p.0; Eample, p. ) Now work the Are You Prepared? problems on page. OBJECTIVES Graph the Functions Listed in the Librar of Functions Graph Piecewise-defined Functions Figure 6 (, ) (, ) 5 (9, ) 0 We now introduce a few more functions to add to our list of important functions. We begin with the square root function. In Section. we graphed the equation =. Figure shows a graph of f =. Based on the graph, we have the following properties: Properties of f(). The -intercept of the graph of f = is 0. The -intercept of the graph of f = is also 0.. The function is neither even nor odd.. It is increasing on the interval 0, q.. It has a minimum value of 0 at = 0. EXAMPLE Graphing the Cube Root Function (a) Determine whether f = is even, odd, or neither. State whether the graph of f is smmetric with respect to the -ais or smmetric with respect to the origin. (b) Determine the intercepts, if an, of the graph of f =. (c) Graph f =.

2 08 CHAPTER Functions and Their Graphs Solution (a) Because f- = - = - = -f the function is odd. The graph of f is smmetric with respect to the origin. (b) The -intercept is f0 = 0 = 0. The -intercept is found b solving the equation f = 0. f = 0 = 0 f() = = 0 Cube both sides of the equation. The -intercept is also 0. (c) We use the function to form Table 6 and obtain some points on the graph. Because of the smmetr with respect to the origin, we onl need to find points, for which Ú 0. Figure shows the graph of f =. Table 6 f() (, ) a 8, b (, ) L.6 A, B 8 (8, ) Figure (, ) 8 (, ) (, ) (, ) (, ) (, ) From the results of Eample and Figure, we have the following properties of the cube root function. 8 Properties of f (). The -intercept of the graph of f = is 0. The -intercept of the graph of f = is also 0.. The function is odd.. It is increasing on the interval - q, q.. It does not have a local minimum or a local maimum. EXAMPLE Graphing the Absolute Value Function (a) Determine whether f = ƒƒ is even, odd, or neither. State whether the graph of f is smmetric with respect to the -ais or smmetric with respect to the origin. (b) Determine the intercepts, if an, of the graph of f = ƒƒ. (c) Graph f = ƒƒ. Solution (a) Because f- = ƒ -ƒ = ƒƒ = f the function is even. The graph of f is smmetric with respect to the -ais.

3 SECTION.5 Librar of Functions; Piecewise-defined Functions 09 (b) The -intercept is f0 = ƒ0ƒ = 0. The -intercept is found b solving the equation f = ƒƒ = 0. So the -intercept is 0. (c) We use the function to form Table 7 and obtain some points on the graph. Because of the smmetr with respect to the -ais, we onl need to find points, for which Ú 0. Figure shows the graph of f = ƒƒ. Table 7 f() ƒƒ (, ) 0 0 (, ) (, ) (, ) Figure (, ) (, ) (, ) (, ) (, ) (, ) From the results of Eample and Figure, we have the following properties of the absolute value function. Properties of f() ƒƒ. The -intercept of the graph of f = ƒƒ is 0. The -intercept of the graph of f = ƒƒ is also 0.. The function is even.. It is decreasing on the interval - q, 0. It is increasing on the interval 0, q.. It has a local minimum of 0 at = 0. Seeing the Concept = ƒƒ Graph on a square screen and compare what ou see with Figure. Note that some graphing calculators use abs for absolute value. Graph the Functions Listed in the Librar of Functions We now provide a summar of the ke functions that we have encountered. In going through this list, pa special attention to the properties of each function, particularl to the shape of each graph. Knowing these graphs will la the foundation for later graphing techniques. Figure Linear Function f( ) m b, m 0 Linear Function f = m + b, m and b are real numbers (0, b) See Figure. The domain of a linear function is the set of all real numbers. The graph of this function is a nonvertical line with slope m and -intercept b. A linear function is increasing if m 7 0, decreasing if m 6 0, and constant if m = 0.

4 0 CHAPTER Functions and Their Graphs Constant Function Figure 5 Constant Function Figure 6 Identit Function b (0,b) f() = b f() = See Figure 5. A constant function is a special linear function m = 0. Its domain is the set of all real numbers; its range is the set consisting of a single number b. Its graph is a horizontal line whose -intercept is b. The constant function is an even function whose graph is constant over its domain. Identit Function f = b, b is a real number f = (, ) Figure 7 Square Function (, ) See Figure 6. The identit function is also a special linear function. Its domain and range are the set of all real numbers. Its graph is a line whose slope is m = and whose -intercept is 0. The line consists of all points for which the -coordinate equals the -coordinate. The identit function is an odd function that is increasing over its domain. Note that the graph bisects quadrants I and III. f() = Square Function (, ) (, ) f = (, ) Figure 8 Cube Function (, ) See Figure 7. The domain of the square function f is the set of all real numbers; its range is the set of nonnegative real numbers. The graph of this function is a parabola whose intercept is at 0, 0. The square function is an even function that is decreasing on the interval - q, 0 and increasing on the interval 0, q. (, ) Figure 9 Square Root Function (, ) f() = f() = (, ) (, ) Cube Function f = See Figure 8. The domain and the range of the cube function is the set of all real numbers.the intercept of the graph is at 0, 0. The cube function is odd and is increasing on the interval - q, q. Square Root Function f = 5 See Figure 9.

5 SECTION.5 Librar of Functions; Piecewise-defined Functions Figure 50 Cube Root Function (, ) 8 (, ) Figure 5 Reciprocal Function (, ) (, ) (, ) (, ) 8 f() = The domain and the range of the square root function is the set of nonnegative real numbers. The intercept of the graph is at 0, 0. The square root function is neither even nor odd and is increasing on the interval 0, q. Cube Root Function f = See Figure 50. The domain and the range of the cube root function is the set of all real numbers.the intercept of the graph is at 0, 0. The cube root function is an odd function that is increasing on the interval - q, q. Reciprocal Function f = (, ) (, ) Refer to Eample, page, for a discussion of the equation = See Figure 5.. The domain and the range of the reciprocal function is the set of all nonzero real numbers. The graph has no intercepts. The reciprocal function is decreasing on the intervals - q, 0 and 0, q and is an odd function. Absolute Value Function Figure 5 Absolute Value Function (, ) (, ) f() = (, ) (, ) f = ƒƒ See Figure 5. The domain of the absolute value function is the set of all real numbers; its range is the set of nonnegative real numbers. The intercept of the graph is at 0, 0. If Ú 0, then f =, and the graph of f is part of the line = ; if 6 0, then f = -, and the graph of f is part of the line = -. The absolute value function is an even function; it is decreasing on the interval - q, 0 and increasing on the interval 0, q. The notation int stands for the largest integer less than or equal to. For eample, int =, int.5 =, inta b = 0, inta - b = -, intp = This tpe of correspondence occurs frequentl enough in mathematics that we give it a name. Greatest Integer Function f = int * = greatest integer less than or equal to * Some books use the notation f = Œœ instead of int.

6 CHAPTER Functions and Their Graphs Table 8 f() int() (, ) (-, - ) a - a - a, 0b a, 0b 0 a, 0b, - b, - b NOTE When graphing a function using a graphing utilit, ou can choose either the connected mode, in which points plotted on the screen are connected, making the graph appear without an breaks, or the dot mode, in which onl the points plotted appear. When graphing the greatest integer function with a graphing utilit, it is necessar to be in the dot mode. This is to prevent the utilit from connecting the dots when f changes from one integer value to the net. See Figure 5. We obtain the graph of f = int b plotting several points. See Table 8. For values of, - 6 0, the value of f = int is -; for values of, 0 6, the value of f is 0. See Figure 5 for the graph. Figure 5 Greatest Integer Function The domain of the greatest integer function is the set of all real numbers; its range is the set of integers. The -intercept of the graph is 0. The -intercepts lie in the interval 0,. The greatest integer function is neither even nor odd. It is constant on ever interval of the form k, k +, for k an integer. In Figure 5, we use a solid dot to indicate, for eample, that at = the value of f is f = ; we use an open circle to illustrate that the function does not assume the value of 0 at =. From the graph of the greatest integer function, we can see wh it also called a step function. At = 0, = ;, = ;, and so on, this function ehibits what is called a discontinuit; that is, at integer values, the graph suddenl steps from one value to another without taking on an of the intermediate values. For eample, to the immediate left of =, the -coordinates are, and to the immediate right of =, the -coordinates are. Figure 5 shows the graph of f = int on a TI-8 Plus. Figure 5 f() = int() 6 6 (a) Connected mode 6 6 (b) Dot mode The functions that we have discussed so far are basic. Whenever ou encounter one of them, ou should see a mental picture of its graph. For eample, if ou encounter the function f =, ou should see in our mind s ee a picture like Figure 7. NOW WORK PROBLEMS 9 THROUGH 6. Graph Piecewise-defined Functions Sometimes a function is defined differentl on different parts of its domain. For eample, the absolute value function f = ƒƒ is actuall defined b two equations: f = if Ú 0 and f = - if 6 0. For convenience, we generall combine these equations into one epression as f = ƒƒ = e if Ú 0 - if 6 0 When functions are defined b more than one equation, the are called piecewisedefined functions. Let s look at another eample of a piecewise-defined function.

7 SECTION.5 Librar of Functions; Piecewise-defined Functions EXAMPLE Analzing a Piecewise-defined Function The function f is defined as f = c (a) Find f0, f, and f. (b) Determine the domain of f. (c) Graph f b hand. (d) Use the graph to find the range of f. - + if - 6 if = if 7 Figure 55 Solution (a) To find f0, we observe that when = 0 the equation for f is given b f = - +. So we have f0 = -0 + = When =, the equation for f is f =. Thus, f = When =, the equation for f is f =. So f = = = 5 (, ) (, ) (, ) (0, ) = + (b) To find the domain of f, we look at its definition. We conclude that the domain of f is 5 ƒ Ú -6, or the interval -, q. (c) To graph f b hand, we graph each piece. First we graph the line = - + and keep onl the part for which - 6. Then we plot the point, because, when =, f =. Finall, we graph the parabola = and keep onl the part for which 7. See Figure 55. (d) From the graph, we conclude that the range of f is 5 ƒ 7 06, or the interval 0, q. To graph a piecewise-defined function on a graphing calculator, we use the TEST menu to enter inequalities that allow us to restrict the domain function. For eample, to graph the function in Eample using a TI-8 Plus graphing calculator, we would enter the function in Y as shown in Figure 56(a). We then graph the function and obtain the result in Figure 56(b). When graphing piecewise-defined functions on a graphing calculator, ou should use dot mode so that the calculator does not attempt to connect the pieces of the function. Figure 56 6 (, ) Y Y (a) (b) NOW WORK PROBLEM 9.

8 CHAPTER Functions and Their Graphs EXAMPLE Cost of Electricit In Ma 00, Commonwealth Edison Compan supplied electricit to residences for a monthl customer charge of $7. plus 8.75 per kilowatt-hour (kwhr) for the first 00 kwhr supplied in the month and 6.08 per kwhr for all usage over 00 kwhr in the month. (a) What is the charge for using 00 kwhr in a month? (b) What is the charge for using 700 kwhr in a month? (c) If C is the monthl charge for kwhr, epress C as a function of. SOURCE: Commonwealth Edison Co., Chicago, Illinois, 00. Solution (a) For 00 kwhr, the charge is $7. plus 8.75 = $ per kwhr. That is, Charge = $7. + $ = $.96 (b) For 700 kwhr, the charge is $7. plus 8.75 per kwhr for the first 00 kwhr plus 6.08 per kwhr for the 00 kwhr in ecess of 00. That is, Charge = $7. + $ $ = $58.85 Figure 57 Charge (dollars) 80 (700, 58.85) (00, 0.) 0 (00,.96) Usage (kwhr) (c) If 0 00, the monthl charge C (in dollars) can be found b multipling times $ and adding the monthl customer charge of $7.. So, if 0 00, then C = For 7 00, the charge is , since - 00 equals the usage in ecess of 00 kwhr, which costs $ per kwhr. That is, if 7 00, then C = The rule for computing C follows two equations: See Figure 57 for the graph. = = if 0 00 C = e if 7 00

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