1.6. Piecewise Functions. LEARN ABOUT the Math. Representing the problem using a graphical model

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1 1. Piecewise Functions YOU WILL NEED graph paper graphing calculator GOAL Understand, interpret, and graph situations that are described b piecewise functions. LEARN ABOUT the Math A cit parking lot uses the following rules to calculate parking fees: A flat rate of $5. for an amount of time up to and including the first hour A flat rate of $1.5 for an amount of time over 1 h and up to and including h A flat rate of $13 plus $3 per hour for each hour after h? How can ou describe the function for parking fees in terms of the number of hours parked? EXAMPLE 1 Representing the problem using a graphical model Use a graphical model to represent the function for parking fees. Solution Time (h) Parking Fee ($) Create a table of values. 1. Piecewise Functions NEL

2 1. Plot the points in the table of values. Use a solid dot to include a value in an interval. Use an open dot to eclude a value from an interval. Cost ($) Time (h) 5 The domain of this piecewise function is $. The function is linear over the domain, but it is discontinuous at 5, 1, and. There is a solid dot at (, ) and an open dot at (, 5) because the parking fee at h is $.. There is a closed dot at (1, 5) and an open dot at (1, 1.5) because the parking fee at 1 h is $5.. There is a closed dot at (, 1.5) and an open dot at (, 13) because the parking fee at h is $1.5. The last part of the graph continues in a straight line since the rate of change is constant after h. piecewise function a function defined b using two or more rules on two or more intervals; as a result, the graph is made up of two or more pieces of similar or different functions Each part of a piecewise function can be described using a specific equation for the interval of the domain. EXAMPLE Representing the problem using an algebraic model Use an algebraic model to represent the function for parking fees. Solution if 5 if, # 1 if 1, # if. Write the relation for each rule. NEL Chapter 1 7

3 f () 5 μ, if 5 5, if, # 1 1.5, if 1, # , if. Combine the relations into a piecewise function. The domain of the function is $. The function is discontinuous at 5, 1, and because there is a break in the function at each of these points. Reflecting A. How do ou sketch the graph of a piecewise function? B. How do ou create the algebraic representation of a piecewise function? C. How do ou determine from a graph or from the algebraic representation of a piecewise function if there are an discontinuities? APPLY the Math EXAMPLE 3 Representing a piecewise function using a graph Graph the following piecewise function.,if, f () 5 e 1 3, if $ Solution Create a table of values. f () 5 f () f() f() From the equations given, the graph consists of part of a parabola that opens up and a line that rises from left to right. Both tables include 5 since this is where the description of the function changes. 1. Piecewise Functions NEL

4 1. 1 = f() Plot the points, and draw the graph. A solid dot is placed at (, 7) since 5 is included with f() An open dot is placed at (, ) since 5 is ecluded from f() 5. f () is discontinuous at 5. EXAMPLE = f () Representing a piecewise function using an algebraic model Determine the algebraic representation of the following piecewise function. Solution 1 f () 5,if #, if. The graph is made up of two pieces. One piece is part of the reciprocal function defined b 5 1 when #. The other piece is a horizontal line defined b 5 when.. The solid dot indicates that point Q, 1 R belongs with the reciprocal function. NEL Chapter 1 9

5 EXAMPLE 5 Reasoning about the continuit of a piecewise function Is this function continuous at the points where it is pieced together? Eplain. 1 1, if # g() 5 1 1, if,, 3, if $ 3 Solution The function is continuous at the points where it is pieced together if the functions being joined have the same -values at these points. Calculate the values of the function at 5 using the relevant equations: () Calculate the values of the function at 5 3 using the relevant equations: (3) The function is discontinuous, since there is a break in the graph at 5 3. The graph is made up of three pieces. One piece is part of an increasing line defined b when #. The second piece is an increasing line defined b when,, 3. The third piece is part of a parabola that opens down, defined b 5 when $ 3. The two -values are the same, so the two linear pieces join each other at 5. The two -values are different, so the second linear piece does not join with the parabola at 5 3. Tech Support For help using a graphing calculator to graph a piecewise function, see Technical Appendi, T-1. Verif b graphing Piecewise Functions NEL

6 1. In Summar Ke Ideas Some functions are represented b two or more pieces. These functions are called piecewise functions. Each piece of a piecewise function is defined for a specific interval in the domain of the function. Need to Know To graph a piecewise function, graph each piece of the function over the given interval. A piecewise function can be either continuous or not. If all the pieces of the function join together at the endpoints of the given intervals, then the function is continuous. Otherwise, it is discontinuous at these values of the domain. CHECK Your Understanding 1. Graph each piecewise function., if, 1 1,if #1 a) f () 5 e d) f () 5 e 3,if $ 1 1, if.1, if, f () 5 e e) 1, if $ 1, if # c) f () 5 e f) f () 5,if, 1,if.,if $ 1. State whether each function in question 1 is continuous or not. If not, state where it is discontinuous. 3. Write the algebraic representation of each piecewise function, using function notation. a) = f () f () 5 e!,if,,if $ = f (). State the domain of each piecewise function in question 3, and comment on the continuit of the function. NEL Chapter 1 51

7 PRACTISING 5. Graph the following piecewise functions. Determine whether each K function is continuous or not, and state the domain and range of the function., if,1 a) f () 5 e c) 3, if $1 f () 5 e 1 1, if, 1 1, if $ 1, if,1 f () 5 e,if # d) f () 5 1, if 1 # # 3,if. 5, if. 3. Graham s long-distance telephone plan includes the first 5 min per A month in the $15. monthl charge. For each minute after 5 min, Graham is charged $.. Write a function that describes Graham s total long-distance charge in terms of the number of long distance minutes he uses in a month. 7. Man income ta sstems are calculated using a tiered method. Under a certain ta law, the first $1 of earnings are subject to a 35% ta; earnings greater than $1 and up to $5 are subject to a 5% ta. An earnings greater than $5 are taed at 55%. Write a piecewise function that models this situation.. Find the value of k that makes the following function continuous. T Graph the function. f () 5 e k,if,1 1, if $1 9. The fish population, in thousands, in a lake at an time,, in ears is modelled b the following function:,if # # f () 5 e 1, if. This function describes a sudden change in the population at time 5, due to a chemical spill. a) Sketch the graph of the piecewise function. Describe the continuit of the function. c) How man fish were killed b the chemical spill? d) At what time did the population recover to the level it was before the chemical spill? e) Describe other events relating to fish populations in a lake that might result in piecewise functions Piecewise Functions NEL

8 1. Create a flow chart that describes how to plot a piecewise function with two pieces. In our flow chart, include how to determine where the function is continuous An absolute value function can be written as a piecewise function that C involves two linear functions. Write the function f () as a piecewise function, and graph our piecewise function to check it. 1. The demand for a new CD is described b 1 D(p) 5 p,if, p # 15, if p. 15 where D is the demand for the CD at price p, in dollars. Determine where the demand function is discontinuous and continuous. Etending 13. Consider a function, f (), that takes an element of its domain and rounds it down to the nearest 1. Thus, f (15.) 5 1, while f (1.7) 5 and f (3) 5 3. Draw the graph, and write the piecewise function. You ma limit the domain to P[, 5). Wh do ou think graphs like this one are often referred to as step functions? 1. Eplain wh there is no value of k that will make the following function continuous. 5, if,1 f () 5 1 k, if 1 # # 3,if The greatest integer function is a step function that is written as f () 5 3, where f () is the greatest integer less than or equal to. In other words, the greatest integer function rounds an number down to the nearest integer. For eample, the greatest integer less than or equal to the number [5.3] is 5, while the greatest integer less than or equal to the number 35.3 is. Sketch the graph of f () a) Create our own piecewise function using three different transformed parent functions. Graph the function ou created in part a). c) Is the function ou created continuous or not? Eplain. d) If the function ou created is not continuous, change the interval or adjust the transformations used as required to change it to a continuous function. NEL Chapter 1 53

9 11. No; several students could have the same grade point average. 1. a) f 1 () 5 1 ( ) 3 h 1 () 5 c) g 1 () 5 " 3 d) m 1 () a) 5 ( 3) 1 1 c) 5 Å 1 d) (., 3.55), (.,.), (3.55,.), (3., 3.) e) $ 3 because a negative square root is undefined. f) g() 5 5, but g 1 (5) 5 or ; the inverse is not a function if this is the domain of g. 1. For 5" 1, D 5 5PR $ and R 5 5PR #. For 5, D 5 5PR and R 5 5PR $. The student would be correct if the domain of 5 is restricted to D 5 5PR #. 15. Yes; the inverse of 5 " 1 is 5 so long as the domain of this second function is restricted to D 5 5PR $. 1. John is correct. Algebraic: ; ; ( ) 5 3 ; 5 " 3 ( ). Numeric: Let Graphical: " 3 (1) 5 " ; 5 " 3 ( ) 5 " 3 (1 ) The graphs are reflections over the line f () 5 k works for all kpr. 5 k Switch variables and solve for : 5 k 5 k So the function is its own inverse. 1. If a horizontal line hits the function in two locations, that means there are two points with equal -values and different -values. When the function is reflected over the line 5 to find the inverse relation, those two points become points with equal -values and different -values, thus violating the definition of a function. Lesson 1., pp a) c) d) e) f). a) Discontinuous at 5 1 Discontinuous at 5 c) Discontinuous at 5 d) Continuous e) Discontinuous at 5 f) Discontinuous at 5 1 and 5 3. a) 3 1 f() 5 e, if # 1 1 1, if. 1, if, 1 f () 5 e!, if $ 1. a) D 5 5PR; the function is discontinuous at 5 1. D 5 5PR; the function is continuous. 5. a) The function is discontinuous at 51. D 5 5PR R 5 5, 3 The function is continuous. D 5 5PR R 5 5f ()PR f () $ 1 Answers NEL

10 c) Answers ma var. For eample: Plot the function for the left interval. Plot the function for the right interval. 1. Answers ma var. For eample: 1 3, if,1 a) f () 5 1 1, if 1 # #! 1 1, if. 5 1 The function is continuous. D 5 5PR R 5 5f ()PR f () $ 1 d) The function is continuous. D 5 5PR R 5 5 f ()PR 1 # f () # 5 15, if # # 5. f () 5 e 15 1., if $ 5 7. f () 5.35, if # # 1.5 1, if 1, # 5.55, if Determine if the plots for the left and right intervals meet at the -value that serves as the common end point for the intervals; if so, the function is continuous at this point. Determine continuit for the two intervals using standard methods. 1 3, if $3 f () e 3, if,3 1. discontinuous at p 5 and p 5 15; continuous at, p, 15 and p c) The function is not continuous. The last two pieces do not have the same value for , if,1 d) f () 5 1 1, if 1 # # 1! 1 1, if. 1 Lesson 1.7, pp a) 5(, ), (, 5), (1, 5), (, 1) 5(, ), (, 3), (1, 1), (, ) c) 5(, ), (, 3), (1, 1), (, ) d) 5(, ), (, ), (1, ), (, ). a) 1. k 5 9. a) The function is discontinuous at 5. c) 3 fish d) 1 5 ; 5 5; 5 1 e) Answers ma var. For eample, three possible events are environmental changes, introduction of a new predator, and increased fishing. 13., if #, 1 1, if 1 #, f () 5 f, if #, 3 3, if 3 #,, if #, It is often referred to as a step function because the graph looks like steps. 1. To make the first two pieces continuous, 5(1) 51 1 k, so k 5. But if k 5, the graph is discontinuous at a) 1 Answers NEL Answers 19