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Project AMP Dr. Antonio R. Quesada Director Project AMP Lesson Plan Introduction to Graphical and Algebraic Representations of Inverses Lesson Summary: The student will complete an inquiry based lesson to discover the need for finding inverses of functions. Emphasis is placed on several real life applications. Special attention is given to using proper function notation and definitions. Key Words: Inverse Inverse Functions Graph Inverses Algebraic Inverses Background Knowledge: Know and understand the definition of a function Use proper function notation Understand function vocabulary including inputs, outputs, and domain A basic understanding of swapping inputs and outputs to generate inverses Recognize reflections over various lines including y = 0, x = 0, y = x Ohio Academic Content Standards: Strand-Patterns, Functions and Algebra Standards Benchmark-Grades 8-10 (C and D) Learning Objectives: Sketch inverses by hand Visualize an inverse given a function Use proper function and inverse notation Generate an algebraic representation of the inverse of a simple function Gain an appreciation for an inverse via a practical application Materials: A graphic calculator Suggested Procedures: Prior to handing out the inquiry based lesson, the teacher must present the basics of finding an inverse given a function. This must include swapping the inputs and outputs of several functions (i.e. 3 f 7 7 f 1 3 ). Attention Getter: Use the DRAW Mode on the graphic calculator to draw inverses. Group Size: 3 or less Assessments: Students must occasionally check answers with the teacher Results will be checked at the end of the activity Project AMP Dr. Antonio R. Quesada Director Project AMP An Introduction to Graphical and Algebraic Representations of Inverses Team Number 12: Michelle Augusta Connie Kramer Michael Scott Kim Snyder Investigation Goals: Sketch inverses by hand Visualize an inverse given a function Use proper function and inverse notation Generate an algebraic representation of the inverse of a simple function Gain an appreciation for an inverse via a practical application Project AMP Dr. Antonio R. Quesada Director Project AMP Solutions to An Introduction to Graphical and Algebraic Representations of Inverses Project AMP Dr. Antonio R. Quesada Director Project AMP An Introduction to Graphical and Algebraic Representations of Inverses 1 As we learned yesterday, if f is a function, then the effect of its inverse, denoted f , is to swap the role of its inputs and outputs. Yesterday we learned the effect of the inverse upon specific input – output combinations. Today we will learn two ways to represent the inverse of a function that will allow us to quickly gain information about any desired input – output combination. First, though, do these exercises to review yesterday’s work. Review Exercises 1 Suppose that both f and f are functions, and that the following are true: 1 f (3) 7 ; f (2) 3 ; f (7) 5 ; f (7 ) 2 If possible, answer the questions that follow. If it is not possible to know the answer to a question using the given information, write “not enough information.” 1 1 1 1) f (5) 2) f ( 7 ) 3) f (3) -7 3 not enough information 1 4) f (2) 5) f (7) 6) f (3) 7 not enough information -2 Graphs of Inverses Complete the following table for the function f ( x) x 6 x 12 x 7 3 2 7) x 0 1 2 3 4 f(x) -7 0 1 2 9 8) With the help of your graphing calculator, sketch the graph of f below. Be sure that the five points listed in the table are included on your graph. Project AMP Dr. Antonio R. Quesada Director Project AMP 9) Using the table you made in #7 and the skills you learned yesterday, answer the following questions. 1 1 1 1 1 (a) f ( 7 ) (b) f (0) (c) f (1) (d) f (2) (e) f (9) 0 1 2 3 4 1 10) Make a table for f whose content reflects the questions and answers from #9 x -7 0 1 2 9 -1 f (x) 0 1 2 3 4 1 11) Make an observation about the relationship between the tables for f and f . You will be asked to use this observation again later. The rows corresponding to input and output have been swapped 1 1 12) Use the five points in the table of f from #10 as a guide to make a sketch of f , connecting the points in a way that makes sense, on the same graph you used to sketch f in #8. 1 Using the processes from #7-12, sketch the graph of g ( x) 2 and its inverse, g . Be sure to complete x 13) both tables below and sketch two graphs, and that each graph contains all points from its respective table. x -1 0 1 2 3 g(x) ½ 1 2 4 8 x ½ 1 2 4 8 -1 g (x) -1 0 1 2 3 Project AMP Dr. Antonio R. Quesada Director Project AMP 14) Looking at the two pairs of graphs you have made of a function with its inverse, make an observation about the geometric relationship between a function and its inverse. [Hint: draw the line y x along with each pair of graphs] The graph of a function and its inverse are symmetric over the line y=x [Depending on their backgrounds, students’ observations may not be so sophisticated] 15) Your calculator has a built – in feature that sketches the inverse of a function. Let’s use this feature to verify our graphs from #8. To get started, type the rule for f ( x) x 6 x 12 x 7 into Y1. Next, 3 2 nd return to the home screen. Press the DRAW button (2 – PRGM). Now you are on the DRAW menu. Either press 8 or scroll down and press ENTER on #8. Either way, you should now see “DrawInv” on your home screen. Do not press ENTER yet. Now you must tell the calculator of what function to draw the inverse. Remember, our function rule is entered into Y 1, so make Y1 appear on the home screen (remember, we do this via the VARS button). Now you should see “DrawInv Y1” on your home screen. Press ENTER. Does it verify your graphs? Yes 16) Repeat the process from #15 to verify the graphs of g and g 1 from #13. I have successfully verified them. Using your calculator exclusively (that is, do not use a table), make a rough sketch of h( x) x 4 x 6 2 17) 1 and h (using DrawInv) in the same xy-plane. Remember, for an undistorted representation, choose a square window by pressing ZOOM:5 (If you change the window after performing DrawInv, you will have to redo the DrawInv) 18) Reread your observation from #14. Is the relationship between the above curves consistent with your observation? If not, adjust your observation below. Answers will vary Project AMP Dr. Antonio R. Quesada Director Project AMP 19) Judging from your sketch of the graph of h 1 , is h 1 a function? Write a sentence to justify your answer. No. It fails the vertical line test. (Alternatively, there exists at least one value of x (say, x=3) with two outputs (namely, 1 and 3)) 20) Looking only at the graph of h (that is, without generating a graph for h 1 ), make a conjecture as to how 1 you could have anticipated that h would not be a function. The existence of a horizontal line along which two points lie for the function h becomes the existence of a -1 vertical line along which two points lie for the relation h Sometimes, the inverse of a function is not, itself, a function. This can lead to some mathematical problems for which there are strategies to handle. We will explore these problems and strategies in a later lesson. A Real – World Example 21) If you are driving a car and need to stop in the shortest distance possible, you will “slam” your brakes. Evidence from your tires will be left on the road of the distance it has taken you to stop, in the form of a skid mark. If two people are driving similarly – sized cars and slam their breaks, will each of their cars create skid marks that are the same length? What do you think is the major factor upon which the length of their respective skid marks depends? No; the length of a set of skid marks will depend chiefly on the speed of the car that generates it. 22) The length of a skid mark that is created by a car depends most on the speed of the car when the break is applied. (Is this what you answered above?) Under normal conditions, the following function m roughly models the length of a skid mark, in feet, generated by a car whose speed is x , in miles per hour: m( x) 0.057 x 2 Using your calculator, generate a graph of this function. Think again about what x represents in this real – world situation. Given this, think about whether any portion of the graph gives information that does not make sense. Give the new, restricted domain of this function by eliminating all values of x that do not make sense (and only those that do not make sense) given what x represents. Justify your response. [If you need a hint, think about what you would be finding if you computed m(5) ] The values of x must be nonnegative because speed must be nonnegative. So the domain is [0, ∞) Project AMP Dr. Antonio R. Quesada Director Project AMP 23) Roughly sketch the graph of m , with its restricted domain, below. (You may need to adjust your window on your calculator…use the units below as a guide) 24) Compute m(40) and write a complete sentence to interpret its meaning (include units with your numbers) 91.2 feet. The skid mark created by a car traveling at a speed of 40 mph whose brakes are slammed will be approximately 91.2 feet. 25) An officer arriving at the scene of an accident measures the skid marks generated by a car to be 65 feet. Why might the function m from above help the officer estimate the speed of the driver when brakes were slammed? Why is it difficult to use the function m directly to find this speed? The function relates speed to length of skid mark, but it takes speed as its input and produces length of skid mark as its output. We do not have a speed to input, so we must work backwards. 26) The function m has its input, x , measure speed in miles per hour and its output, m(x) , measure length of skid mark in feet. It would be much more convenient for us to have a function that swaps these roles. a) If we made a new function that swapped these roles, what would the input to this new function measure? What would its output measure? Input: length of skid mark, in feet Output: speed of car when brakes are slammed, in miles per hour b) What is the name for the mathematical concept that will have this effect on a function? What will we symbolically call our new function in this particular case? -1 Inverse; m Project AMP Dr. Antonio R. Quesada Director Project AMP 27) As you learned to earlier, use the DrawInv feature on your calculator to draw the inverse of m . Given the restriction that we had to impose on m in #22, explain why the entire graph generated by DrawInv is not the inverse of m . DrawInv drew the inverse of the portion of the function that was restricted, as well. 28) Which portion of the graph generated by DrawInv should be discarded? Justify your answer using what was learned earlier about how the graph of an inverse is generated. The portion of the graph below the x-axis (with negative y-values), as this corresponds to the portion of the domain that was restricted (with negative x-values). 1 29) Sketch the graph of m below, taking into account your answer from #27. Be sure to label your axes with the correct units. (You may have to adjust your viewing window) 30) Using your sketch from #29, roughly estimate the speed of the car that generated the skid mark of 65 feet. Explain the procedure you used to arrive at this estimate. Approximately 34 feet; by drawing the vertical line x = 65 and drawing a perpendicular from its point of intersection with the graph of m to the y-axis. 31) Do you trust the accuracy of your estimate from #30 to, say, be within one-hundredth (.01) of a mile per hour? What makes it difficult to obtain such accuracy using a graph? No; the above graphical method has too much inexactness associated with it, from our sketch of the graph to the thickness of the curves drawn. Algebraic Representations of Inverses 32) Let’s try to get a more exact answer to the question from #30. Remember, for the function m , the input, x , represents a speed in miles per hour and the output, m(x) represents a skid mark length, in feet. If we know that the length of a skid mark produced by a car is 65 feet, which variable do we know the value of, x or m(x) ? What is the value of this variable? m(x); m(x) = 65 Project AMP Dr. Antonio R. Quesada Director Project AMP 33) The algebraic representation of m is the equation from above that relates the variables x and m(x) . Remember that m( x) 0.057 x . This equation has two variables, x and m(x) . Using your answer 2 from #32, remove the variable you know the value of, replace it with its value, and write your new equation below. 2 65 = 0.057x 34) We have learned strategies for solving the type of equation that you wrote down in #33. Showing your algebraic work, solve this equation. Use your calculator to round your final answer to the nearest hundredth. x≈ 33.77 35) Algebraically, your equation from #33-34 should have had two solutions. (Did you get two solutions?) Below, state the two solutions, tell which solution should be discarded and explain why. 33.77 or -33.77. -33.77 should be discarded, as it is not feasible as a speed (speed cannot be negative) -1 (Alternatively, we discarded this point as a point on the graph of m earlier) 36) Repeat the processes from above to find the speed, to the nearest hundredth of a mile per hour, of cars that make skid marks of 36 feet and 117 feet. Show your setups and algebraic work. 2 2 36 = 0.057x 117 = 0.057x x ≈ 25.13 miles per hour x ≈ 45.31 miles per hour 37) Each time you find a speed, notice that you algebraically solve for x . Go back to the original algebraic representation for m , m( x) 0.057 x . Instead of replacing m(x) with a specific value, as you did in 2 #34 and #36, let’s leave the equation as it is, but still solve for x . [Hint: The only difference will be that, m(x ) after the first step, you won’t be able to compute the left side, it will have to remain ]. .057 m(x) x= 0.057 38) Just as in #34 and #36, the equation above has two solutions. Explain why it was correct to discard one of the solutions. Because x represents speed, and speed cannot be negative (we discarded negative values of speed earlier) Project AMP Dr. Antonio R. Quesada Director Project AMP 39) You should now have an equation that is solved for x , the speed in miles per hour, as a function of m(x) , the length of the skid mark, in feet. Check your equation with me before continuing. 40) The equation from #39 is a function, whose input is length of skid mark in feet, and whose output is speed in miles per hour. It does not look like a function the way we normally see functions, because the variables are not assigned as normal. For instance, normally, the input is assigned the variable name x . To begin to correct this problem, rewrite your equation from #39, but remove the variable names x and m(x) . Replace them with sets of parentheses. (___) ( )= 0.057 41) Take your equation from #39 and insert x into the parentheses that were previously occupied by m(x) . This renames the input using the normal input variable, x . Into the parentheses that were previously 1 1 occupied by x , insert m ( x ) . State what the variables x and m ( x ) represent (include their units). -1 x Equation: m (x) = 0.057 x: length of skid marks, in feet m 1 ( x ) : speed of car when brakes are slammed, in miles per hour 1 We could have given any name to our new function, but we used m to remind ourselves that this function is the inverse of m . Notice that in #41 we swapped the roles of input and output, the same procedure we learned yesterday and used earlier when we were graphing inverses. 42) On your calculator, generate a graph of your function from #41. How does it differ from the graph you obtained by performing the DrawInv feature on the original function m ? It only contains the portion of the graph on and above the x-axis Extend: Try to explain this difference. [Hint: See your answer from #38] DrawInv includes both solutions to the equation from #38, the inverse of the function with unrestricted domain. The function from #41 is the inverse of m with its restricted domain. 1 43) Now that we have an algebraic representation of m , we can use this to quickly and accurately find the length of the skid mark generated by a car traveling at a given speed. Use your algebraic representation of m 1 from #41 to find the following to the nearest hundredth of a mile per hour: (a) m 1 (65) (b) m 1 (36 ) (c) m 1 (117 ) 33.77 mph 25.13 mph 45.31 mph Project AMP Dr. Antonio R. Quesada Director Project AMP (d) Do your answers for (a) – (c) agree with their respective counterparts from #35 – 36? Yes 1 1 44) To show that you understand what the functions m and m represent, compute m(50) and m (50 ) to the nearest hundredth, and write a complete sentence to interpret the meaning of each (include the units of the numbers in your sentences). m(50) 142.5 feet Sentence: A car traveling at a speed of 50 mph at the moment its brakes are slammed will produce a skid mark of approximately 142.5 feet. m 1 (50 ) 29.62 miles per hour Sentence: A car that produced a skid mark of 50 feet was traveling approximately 29.62 miles per hour at the moment its brakes were slammed. 45) Find an algebraic representation for the inverse of the following functions using the procedure from #37 - #41. Show each step of your work. (Note that, unlike the example done in #37 - #41, this process does not always result in two solutions; neither of the examples below have two solutions). (a) f ( x) 3x 12 (b) g ( x) x 3 5 x 12 -1 f (x) = -1 g (x) = 3 x5 3 46) Using the skills you learned earlier in the lesson, try to think of a method you could use to verify that your answers from #45 are correct. 47) One way to convince yourself that your answers from #45 are correct is to use the DrawInv feature on your calculator (is this what you thought of?). Type the algebraic rule for f from #45a into Y1. Draw its inverse 1 using DrawInv. Now, type the algebraic rule you found for f into Y2. If your answer is correct, it should generate the same graph that was generated by DrawInv. Are you confident that your answer from #45a is correct? Repeat this process for #45b. Are you confident that it is correct? (If not, try to find your algebraic errors and repeat this process until you are confident in their correctness) Answers will vary, but, hopefully, most will be “Yes”