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1 What if it is in pieces? Piecewise Functions and an Intuitive Idea of Continuity Teacher Version Lesson Objective: Length of Activity: Students will: Recognize piecewise functions and the notation used to express them. Reinforce the idea of piecewise functions by learning how to graph them on their calculators. Develop an informal definition of continuity. One day (approximately 50 minutes) Core Problems: Problems 1 through 6 Where does this go? Materials: Suggested Lesson Activity: Students need a strong foundation in all types of functions for the work to come in this and future math courses, especially calculus. This introduction to piecewise functions facilitates the students ability to work with multiple functions simultaneously. Additionally, the intuitive idea of continuity prepares students for the formal definition in later courses. The beginning unit analysis lays groundwork for making sense of the area under a curve, a key idea of calculus. Graphing calculators Transparencies (optional) We begin this lesson with a brief overview of functions and a historical note regarding how the concept of a function has been expanded over the years. If students ask, the modern definition of a function has no requirement that there be any continuity whatsoever. The classic example of this is a function which is 1 when x is irrational and 0 when x is rational. In problems 1 and 2, students are introduced to piecewise functions and the notation used to express them. Have students investigate this piecewise function as they have done in the past with other functions. Briefly have the teams report on their findings in problem 1 before beginning problem 2. You may want to supply the teams with transparencies on which to sketch the piecewise function graph so that it can be better shared. Students may struggle some with writing the function for the second part of the piecewise function in problem 2. Remind them of pointslope form to help them with this. A pay rate of $6/hr is the slope and (5, 20) is a point on the second piece. If students continue to struggle with this, suggest that they graph the function.
2 Closure: Problems 3 and 4 introduce another piecewise function and explain how the graphing calculator can be used to graph piecewise functions. Be aware that different model calculators may graph piecewise functions with a vertical line connecting the two pieces of the function. That line should not be present. Show students how to avoid this problem by graphing the function in dot mode. You can either set the calculator to graph in dot mode from the mode key, or by moving the cursor to the left of Y1 and pressing enter until the icon changes to dots. Students may be confused by the notation used by the calculator to graph piecewise functions, since it appears to be multiplying and adding functions together. You may want to explain that the calculator sets the conditions X 1 and X > 1 to a value of 1 when the condition is true, and a value of 0 when the condition is false. In this way the calculator includes the proper piece of the function for the proper xvalues by multiplying that piece by either 0 or 1. Problems 5 and 6 introduce the concept of continuity, relate the concept to the piecewise function in problem 4, and supply another example of a piecewise function that is also discontinuous. In this lesson, we just want students to understand continuity intuitively. Resist the urge to define it more precisely at this point. Be sure that students understand the first homework problem (Kristof s Earnings), as it lays important groundwork for understanding units in calculus. Verify that students have a firm grasp of piecewise functions, what they are, how to represent them algebraically, what applications they can model, and (less importantly, but useful) how to graph them using their calculators. If time permits, you might draw some arbitrary graphs on the board (piecewise and others), and ask students to tell you if the graphs are functions, if they are continuous, and why or why not. Homework: Problems 7 through 14
3 What if it is in pieces? Piecewise Functions and an Intuitive Idea of Continuity By now you have become quite familiar with functions. We have defined them to be relationships that assign one unique output for any given input. You have worked extensively with common functions including lines, parabolas, and exponentials. You also understand that circles and parabolas, which open to the right or left, are not functions because they fail the vertical line test. Now get ready to learn about a slightly different kind of function, one that comes in pieces. This kind of function is known as a piecewise function. A Historical Perspective on Piecewise Functions: A More General Notion for a Function The idea of a function, which is so central to modern mathematics, took a long time to be made precise. For Newton and Leibniz, the originators of calculus in the late 1600 s, functions were simply expressible by a single equation and were always continuous. (Do not worry, we will talk more about what continuous means shortly.) As calculus evolved, questions arose which forced mathematicians to develop ever more general ideas about functions. By the mid 1700 s, Euler (pronounced Oiler ), Daniel Bernoulli, and d Alembert had agreed that what we now call a piecewise function was acceptable. It was not until 1820 that Fourier (pronounced FOOREE AY) had made the step to the current, very general notion of a function. M ATH N OTES Piecewise Function STOP A PIECEWISE FUNCTION is defined as a function that uses different rules for different parts of its domain. Example: 2x 5 for x < 3 Let f (x) = (x 3) for x 3 Note: We will refer to the xvalue where the function changes as the transition point.
4 1. Let f (x) = x2 + 2 for x x + 1 for x > 2. With your study team, investigate this function thoroughly just as you have investigated any other function in the past. Remember all the aspects of investigating: Make a careful sketch, state the domain and range, and identify any x and/or yintercepts. Does this function have a parent graph? Can you identify any symmetry or asymptotes? Once you and your teammates have agreed, be ready to share your investigation with the rest of the class. [ See graph at right above. ] 2. Let us look at an application of a piecewise function. Lisa makes $4/hr babysitting before midnight and $6/hr after midnight. She begins her job at 7 PM. [ a: See table below. b: 4t; 6(t 5) + 20 ] a. Complete the table below for the total amount of money Lisa makes. Time 8PM 9PM 10PM 11:30 12:00 12:30AM 1AM 2AM Hours [ 1 ] 2 3 [ 4.5 ] 5 [ 5.5 ] 6 7 Sitting Amount Earned $4 [ $8 ] [ $12 ] [ $18 ] [ $20 ] [ $23 ] [ $26 ] [ $32 ] b. If we want to fill out the entries after midnight in the table above, we need to realize that the function is piecewise; that is, Lisa is paid at two different rates, one for the time she babysits before midnight, and another for the time she babysits after midnight. Since the rate changes at t = 5, we need two different rules: one for t 5 and one for t > 5. Find the two different functions that would define how much money Lisa makes with respect to the number of hours she babysits. Express the function as a piecewise function using the notation shown in the previous problem. Use the notation shown below to get started. f (t) =... for 0 t 5... for t > 5
5 3. Here is another piecewise function. [ a: See table below. b: See graph at right below. Closed at (1, 3), open at (1, 9). c: D = (, ); R = [2, ), Note: Interval notation is not required in students answers. ] x for x 1 f (x) = 2x + 7 for x > 1 a. Fill out the table for y below. For x 1 x For x > 1 y = x2 + 2 [ 27 ] [ 18 ] [ 11 ] [6] [3] [2] [3] x y = 2x + 7 [9] [ 11 ] [ 13 ] [ 15 ] [ 17 ] [ 19 ] [ 21 ] b. Using your table, make a careful sketch of the graph y = f (x). Recall the use of open circles to indicate that an end point is not included and closed circles to indicate that an end point is included. At which points will the open and closed circles be located on this graph? c. What are the domain and the range of this function? USING TECHNOLOGY TO GRAPH PIECEWISE FUNCTIONS Now that you have experienced graphing a piecewise function by hand, let us take a look at how your calculator can help. You can enter piecewise functions into your calculator by carefully defining each piece in the following way: Enter in Y1: Y1 = (X^2 + 2) ( X 1) + (2X + 7) (X > 1) To get the best view of this function, set your window carefully based on your previous sketch or on the table above. Verify that what you graphed by hand is the same as the graph on the calculator screen.
6 5. We can use the piecewise function you just graphed to investigate an important concept in calculus, the idea of continuity. Read the following Math Note and use it to help answer the questions below. M ATH N OTES Intuitive Notion of Continuity STOP We will formally define continuity later in the course. For now, we can say a function is CONTINUOUS if you can draw the graph of the function without lifting your pencil from the paper. Here are graphs of two continuous functions. Explore the function you graphed on your calculator for problem 4. Does it appear to be broken at a certain point? Could you trace the function using your pencil, without picking the pencil up? As our intuitive definition states, since you have to pick up your pencil to bridge the gap in the function, this function is not continuous. We say that the function is not continuous at one value of x. At which value of x do you think this function is not continuous? [ x = 1 ] 6. Sketch or use your calculator to make a graph of the piecewise function g(x). [ a: D = (, ); R = (, 5) ; b: No, jump at x = 2. ] 2 x for x 2 g(x) = 2x + 9 for x > 2 a. Find the domain and range of the function. b. Is this equation continuous for all values of the domain? Explain how you know.
7 7. KRISTOF S EARNINGS Kristof worked at MacDonut s for $8.00/hr. One Friday night he worked 5 hours. [ a: $40, $8/hr; b: See graph at right below. horizontal; c: 5; d: Each rectangle represents eight dollars of Kristof s earnings, ($8 /hour) (hours) = $8. ] a. How much did he earn that night? What is his rate of pay? b. Make a very careful graph of Kristof s pay rate on a grid where the horizontal axis represents hours and the vertical axis represents dollars earned per hour. Because Kristof s pay rate is constant, it will be represented by what kind of line? c. The area under your graph should be a rectangle. Divide the rectangle you get into smaller rectangles by drawing vertical lines at the 1hour, 2hour, 3hour, 4hour, and 5hour marks. How many rectangles have your created? d. Explain to the other people on your team what the area of each of the rectangles you created in part (c) represents. Hint: What are the units of each side? 8. If f (x) = 3x 2 + 5x 2, show that f (x + 2) = 3x x If f (x) = 3x 2 + 5x 2, show that f (x + h) = 3x 2 + x(6h + 5) + (3h 2 + 5h 2). 10. Factor out (x + 2) from each expression, then simplify. [ a: (x + 2)(x 1), b: x(x + 2)(x + 4), c: 2(x + 2)(x + 5) ] a. (x + 2) 2 3(x + 2) b. (x + 2) 3 4(x + 2) c. 2(x + 2) 2 + 6(x + 2)
8 11. Solve for x in the simplest manner (no calculator or Quadratic Formula). [ a: x = 0 or 3, b: x = 3 or 9 ] a. 5x 2 = 15x b. 2(x 6) = Salvador prefers to evaluate 81 3/4 in a different way. He knows that 3 4 = 81, so he substitutes and uses the power law for exponents: 81 3/4 = (3 4 ) 3/4 = 3 3 = 27. Use Salvador's technique to evaluate each of these. [ a: 1 27, b: 9, c: 25 16, d: 3 ] a b c. 125 ( 64 ) 2 3 d. 1 ( 9 ) 1 2 Hint: Express 1 9 as a power of Let A = ( 2, 5) and B = (5, 2). [ a: 98 = 7 2 ; b: y 5 = (x + 2), y + 2 = (x 5), y = x + 3 ] a. Find the distance between A and B. b. Write the equation of the line passing through points A and B in pointslope form and slopeintercept form. 3 + x 2 if x < Let g(x) = 2x if 2 x <1. Complete parts a f below. 11 x 2 if x 1 [ a: 2, b: 2, c: 4, d: 5, e: See graph at right. f: D and R = all reals ] a. g(3) b. g( 1) c. g( 2) d. g(4) e. Graph g(x). f. State the domain and range of g(x).
9 What if it is in pieces? Piecewise Functions and an Intuitive Idea of Continuity By now you have become quite familiar with functions. We have defined them to be relationships that assign one unique output for any given input. You have worked extensively with common functions including lines, parabolas, and exponentials. You also understand that circles and parabolas, which open to the right or left, are not functions because they fail the vertical line test. Now get ready to learn about a slightly different kind of function, one that comes in pieces. This kind of function is known as a piecewise function. A Historical Perspective on Piecewise Functions: A More General Notion for a Function The idea of a function, which is so central to modern mathematics, took a long time to be made precise. For Newton and Leibniz, the originators of calculus in the late 1600 s, functions were simply expressible by a single equation and were always continuous. (Do not worry, we will talk more about what continuous means shortly.) As calculus evolved, questions arose which forced mathematicians to develop ever more general ideas about functions. By the mid 1700 s, Euler (pronounced Oiler ), Daniel Bernoulli, and d Alembert had agreed that what we now call a piecewise function was acceptable. It was not until 1820 that Fourier (pronounced FOOREE AY) had made the step to the current, very general notion of a function. M ATH N OTES Piecewise Function STOP A PIECEWISE FUNCTION is defined as a function that uses different rules for different parts of its domain. Example: 2x 5 for x < 3 Let f (x) = (x 3) for x 3 Note: We will refer to the xvalue where the function changes as the transition point.
10 1. Let f (x) = x2 + 2 for x x + 1 for x > 2. With your study team, investigate this function thoroughly just as you have investigated any other function in the past. Remember all the aspects of investigating: Make a careful sketch, state the domain and range, and identify any x and/or yintercepts. Does this function have a parent graph? Can you identify any symmetry or asymptotes? Once you and your teammates have agreed, be ready to share your investigation with the rest of the class. 2. Let us look at an application of a piecewise function. Lisa makes $4/hr babysitting before midnight and $6/hr after midnight. She begins her job at 7 PM. a. Complete the table below for the total amount of money Lisa makes. Time 8PM 9PM 10PM 11:30 12:00 12:30AM 1AM 2AM Hours Sitting Amount Earned $4 b. If we want to fill out the entries after midnight in the table above, we need to realize that the function is piecewise; that is, Lisa is paid at two different rates, one for the time she babysits before midnight, and another for the time she babysits after midnight. Since the rate changes at t = 5, we need two different rules: one for t 5 and one for t > 5. Find the two different functions that would define how much money Lisa makes with respect to the number of hours she babysits. Express the function as a piecewise function using the notation shown in the previous problem. Use the notation shown below to get started. f (t) =... for 0 t 5... for t > 5
11 3. Here is another piecewise function x2 + 2 f (x) = 2x + 7 a. for x 1 for x > 1 Fill out the table for y below. For x 1 x y = x2 + 2 For x > 1 x y = 2x + 7 b. Using your table, make a careful sketch of the graph y = f (x). Recall the use of open circles to indicate that an end point is not included and closed circles to indicate that an end point is included. At which points will the open and closed circles be located on this graph? c. What are the domain and the range of this function? USING TECHNOLOGY TO GRAPH PIECEWISE FUNCTIONS Now that you have experienced graphing a piecewise function by hand, let us take a look at how your calculator can help. You can enter piecewise functions into your calculator by carefully defining each piece in the following way: Enter in Y1: Y1 = (X^2 + 2) ( X 1) + (2X + 7) (X > 1) To get the best view of this function, set your window carefully based on your previous sketch or on the table above. Verify that what you graphed by hand is the same as the graph on the calculator screen.
12 5. We can use the piecewise function you just graphed to investigate an important concept in calculus, the idea of continuity. Read the following Math Note and use it to help answer the questions below. M ATH N OTES Intuitive Notion of Continuity STOP We will formally define continuity later in the course. For now, we can say a function is CONTINUOUS if you can draw the graph of the function without lifting your pencil from the paper. Here are graphs of two continuous functions. Explore the function you graphed on your calculator for problem 4. Does it appear to be broken at a certain point? Could you trace the function using your pencil, without picking the pencil up? As our intuitive definition states, since you have to pick up your pencil to bridge the gap in the function, this function is not continuous. We say that the function is not continuous at one value of x. At which value of x do you think this function is not continuous? 6. Sketch or use your calculator to make a graph of the piecewise function g(x). 2 x for x 2 g(x) = 2x + 9 for x > 2 a. Find the domain and range of the function. b. Is this equation continuous for all values of the domain? Explain how you know.
13 7. KRISTOF S EARNINGS Kristof worked at MacDonut s for $8.00/hr. One Friday night, he worked 5 hours. a. How much did he earn that night? What is his rate of pay? b. Make a very careful graph of Kristof's pay rate on a grid where the horizontal axis represents hours and the vertical axis represents dollars earned per hour. Because Kristof s pay rate is constant, it will be represented by what kind of line? c. The area under your graph should be a rectangle. Divide the rectangle you get into smaller rectangles by drawing vertical lines at the 1hour, 2hour, 3hour, 4hour, and 5hour marks. How many rectangles have your created? d. Explain to the other people on your team what the area of each of the rectangles you created in part (c) represents. Hint: What are the units of each side? 8. If f (x) = 3x 2 + 5x 2, show that f (x + 2) = 3x x If f (x) = 3x 2 + 5x 2, show that f (x + h) = 3x 2 + x(6h + 5) + (3h 2 + 5h 2). 10. Factor out (x + 2) from each expression, then simplify. a. (x + 2) 2 3(x + 2) b. (x + 2) 3 4(x + 2) c. 2(x + 2) 2 + 6(x + 2)
14 11. Solve for x in the simplest manner (no calculator or Quadratic Formula). a. 5x 2 = 15x b. 2(x 6) = Salvador prefers to evaluate 81 3/4 in a different way. He knows that 3 4 = 81, so he substitutes and uses the power law for exponents: 81 3/4 = (3 4 ) 3/4 = 3 3 = 27. Use Salvador's technique to evaluate each of these. a b c. 125 ( 64 ) 2 3 d. 1 ( 9 ) 1 2 Hint: Express 1 9 as a power of Let A = ( 2, 5) and B = (5, 2). a. Find the distance between A and B. b. Write the equation of the line passing through points A and B in pointslope form and slopeintercept form. 3 + x 2 if x < Let g(x) = 2x if 2 x <1. Complete parts a f below: 11 x 2 if x 1 a. g(3) b. g( 1) c. g( 2) d. g(4) e. Graph g(x). f. State the domain and range of g(x).
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