11.1 Inverses of Simple Quadratic and Cubic Functions

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1 Locker LESSON 11.1 Inverses of Simple Quadratic and Cubic Functions Teas Math Standards The student is epected to: A..B Graph and write the inverse of a function using notation such as f (). Also A..A, A..C, A.7.I Mathematical Processes A.1.F The student is epected to analze mathematical relationships to connect and communicate mathematical ideas. Language Objective 1.F,.C.1,.C.,.I.5.D,.G. Fill in an organizer of quadratic and cubic functions and their inverses. ENGAGE Essential Question: What functions are the inverses of quadratic and cubic functions, and how can ou find them? Possible answer: Square root functions are the inverses of quadratic functions, with the included restricted domain. Cube root functions are the inverses of cubic functions. You can find the inverse functions b using inverse operations and switching the variables, but must restrict the domain of a quadratic function. PREVIEW: LESSON PERFORMANCE TASK View the Engage section online. Discuss the photo and how the irrigation area can be described b a quadratic function. Ask students to list the variables involved in the function. Then preview the Lesson Performance Task. Name Class Date 11.1 Inverses of Simple Quadratic and Cubic Functions Essential Question: What functions are the inverses of quadratic functions and cubic functions, and how can ou find them? A..B Graph and write the inverse of a function using notation such as f (). Also A..A, A..C, A.7.I Eplore Finding the Inverse of a Man-to-One Function The function ƒ () is defined b the following ordered pairs: (-, ), (, ), (0, 0), (1, ), and (, ). Find the inverse function of ƒ (), ƒ (), b reversing the coordinates in the ordered pairs. (, -), (, ), (0, 0), (, 1), (, ) Is the inverse also a function? Eplain. No; the -values and are both paired with more than one -value. If necessar, restrict the domain of ƒ () such that the inverse, ƒ (), is a function. To restrict the domain of ƒ () so that its inverse is a function, ou can restrict it to 0 With the restricted domain of ƒ (), what ordered pairs define the inverse function ƒ ()? (0, 0), (, 1), (, ) Reflect Resource Locker 1. Discussion Look again at the ordered pairs that define ƒ (). Without interchanging the coordinates, how could ou have known that the inverse of ƒ () would not be a function? When two or more ordered pairs have the same -value but unique -values, the inverse of that function will have ordered pairs in which the same -value maps to more than one -value, and thus will not be a function. That is, if a function is man-to-one, its inverse will not be a function.. How will restricting the domain of ƒ () affect the range of its inverse? Since the domain of a function is the same as the range of its inverse, restricting the domain of f () restricts the range of its inverse. Module Lesson 1 Name Class Date 11.1 Inverses of Simple Quadratic and Cubic Functions Essential Question: What functions are the inverses of quadratic functions and cubic functions, and how can ou find them? A..B Graph and write the inverse of a function using notation such as f (). Also A..A, A..C, A.7.I Eplore Finding the Inverse of a Man-to-One Function The function ƒ () is defined b the following ordered pairs: (-, ), (, ), (0, 0), (1, ), and (, ). Find the inverse function of ƒ (), ƒ (), b reversing the coordinates in the ordered pairs. (, -), (, ), (0, 0), (, 1), (, ) Is the inverse also a function? Eplain. No; the -values and are both paired with more than one -value. If necessar, restrict the domain of ƒ () such that the inverse, ƒ (), is a function. To restrict the domain of ƒ () so that its inverse is a function, ou can restrict it to With the restricted domain of ƒ (), what ordered pairs define the inverse function ƒ ()? Reflect 0 (0, 0), (, 1), (, ) Resource 1. Discussion Look again at the ordered pairs that define ƒ (). Without interchanging the coordinates, how could ou have known that the inverse of ƒ () would not be a function? When two or more ordered pairs have the same -value but unique -values, the inverse of that function will have ordered pairs in which the same -value maps to more than one -value, and thus will not be a function. That is, if a function is man-to-one, its inverse will not be a function.. How will restricting the domain of ƒ () affect the range of its inverse? Since the domain of a function is the same as the range of its inverse, restricting the domain of f () restricts the range of its inverse. Module Lesson 1 HARDCOVER PAGES 07 1 Turn to these pages to find this lesson in the hardcover student edition. 571 Lesson 11.1

2 Eplain 1 Finding and Graphing the Inverse of a Simple Quadratic Function The function ƒ () = is a man-to-one function, so its domain must be restricted in order to find its inverse function. If the domain is restricted to 0, then the inverse function is ƒ () = _ ; if the domain is restricted to 0, then the inverse function is ƒ () = - _. The inverse of a quadratic function is a square root function, which is a function whose rule involves _. The parent square root function is g () = _. A square root function is defined onl for values of that make the epression under the radical sign nonnegative. Eample 1 ƒ () = 0.5 Restrict the domain of each quadratic function and find its inverse. Confirm the inverse relationship using composition. Graph the function and its inverse. Restrict the domain. 0 Find the inverse. EXPLORE Finding the Inverse of a Man-to-One Function INTEGRATE TECHNOLOGY Students can use the DrawInv capabilit of their graphing calculators to draw the inverse of the graph of this function and others, but the should realize that the calculator will draw the inverse whether or not the inverse is also a function. Replace ƒ () with. = 0.5 Multipl both sides b. = Use the definition of positive square root. Switch and to write the inverse. Replace with ƒ (). Confirm the inverse relationship using composition _ = _ = ƒ () = _ ƒ (ƒ()) = ƒ (0.5 ) ƒ(ƒ ()) = 0.5 ( _ ) = (0.5 ) = 0.5() = _ = for 0 = for 0 Since ƒ (ƒ () ) = for 0 and ƒ( ƒ ()) = for 0, it has been confirmed that ƒ () = _ for 0 is the inverse function of ƒ () = 0.5 for 0. Graph ƒ () b graphing ƒ () over the restricted domain and reflecting the graph over the line =. QUESTIONING STRATEGIES How can ou tell if a set of ordered pairs that represents a function has an inverse that is a function? Check to see whether each ordered pair has a unique -value. EXPLAIN 1 Finding and Graphing the Inverse of a Simple Quadratic Function QUESTIONING STRATEGIES What characteristic of the graph of a quadratic function tells ou that the domain needs to be restricted for the inverse to be a function? The U-shape tells ou that the function is not one-to-one, and that its inverse will not be a function unless the domain is restricted. Module Lesson 1 PROFESSIONAL DEVELOPMENT Math Background The domain of an even function f () = n where n =,,, needs to be restricted for the inverse to also be a function. This is because these functions are not one-to-one. If the domain is not restricted, then reflecting its graph across the line = ields a graph that fails the vertical line test. This is because ever non-zero real number has two nth roots when n is even, one positive and one negative. In contrast, the domain of an odd function f () = n where n = 1,, 5, need not be restricted for the inverse to also be a function. This is because ever real number has a single nth root when n is odd, either positive, negative, or zero at the origin. How do ou find the domain and range of the inverse of a quadratic function? The domain will be the range of the original quadratic function. The range will be the domain of the original quadratic function, restricted to values either greater than or equal to, or less than or equal to, the value of that produces the minimum or maimum value of the original function. Inverses of Simple Quadratic and Cubic Functions 57

3 INTEGRATE MATHEMATICAL PROCESSES Focus on Math Connections Remind students how the vertical line test can be used to determine whether the graph of a relation represents a function. Help them see that the horizontal line test can be used to see whether a function has an inverse that is a function. Lead them to recognize how appling the horizontal line test to the graph of the original function is related to appling the vertical line test to the graph of the inverse of the function. B ƒ () = - 7 Restrict the domain. Find the inverse. Replace ƒ () with. = - 7 Add 7 to both sides. + 7 = Use the definition of positive square root. Switch and to write the inverse. Replace with ƒ (). Identif the domain of ƒ (). -7 Confirm the inverse relationship using composition. + 7 = ƒ (ƒ()) = ƒ ( ) ƒ ( ƒ ()) = ƒ ( ) = = = = + 7 = + 7 = f () ( - 7) + 7 ( ) = for 0 = for - 7 Since ƒ (ƒ () ) = for and ƒ ( ) = for, it has been confirmed that ƒ () = for is the inverse function of ƒ () = - 7 for. 0 ƒ () Graph ƒ () b graphing ƒ () over the restricted domain and reflecting the graph over the line = Module Lesson 1 COLLABORATIVE LEARNING Peer-to-Peer Activit Have students work in pairs. Provide each pair with a function of the form f () = a + c. Have students find f (), the inverse of the function. Then have them graph both f () and f () on the same coordinate grid. Have them check their work b verifing that if (, ) belongs to f (), then (, ) belongs to f (). 57 Lesson 11.1

4 Your Turn Restrict the domain of each quadratic function and find its inverse. Confirm the inverse relationship using composition. Graph the function and its inverse.. ƒ () = Restrict the domain. { 0}. Find the inverse. Replace f () with. = Divide both sides b. Use the definition of positive square root. Switch and to write the inverse. Replace with f (). = = = = f () AVOID COMMON ERRORS Students ma make algebraic errors when finding the inverse of a function such as f () = 5. Reinforce that students should isolate before taking the square root of each side, and that when taking the square root, the must take the square root of the entire epression on the other side of the equation, not just of the variable. Confirm the inverse relationship using composition. f (f()) = f ( ) = = = for 0 Since f (f () ) = for 0, it has been confirmed that f () = _ for 0 is the inverse function of f () = for 0. Graph f () b graphing f () and reflecting f () over the line =.. ƒ () = 5 Restrict the domain { 0}. Find the inverse. Replace f () with. = 5 Divide both sides b 5. Use the definition of postive square root. Switch and to write the inverse. Replace with f (). _ 5 = _ 5 = _ 5 = _ 5 = f () Confirm the inverse relationship using composition. f (f () ) = f ( 5 ) = _ 5 = 5 = for 0 Since f (f() ) = for 0, it has been confirmed that f () = _ function of f () = 5 for 0. Graph f () b graphing f () and reflecting f () over the line =. 0 for 0 is the inverse 5 Module Lesson 1 DIFFERENTIATE INSTRUCTION Graphic Organizers Students ma benefit from summarizing what the ve learned in a graphic organizer like the one shown below. Encourage students to add graphs. Functions Linear Quadratic Cubic f () = a f () = a f () = a f () = _ a f () = _ a f = _ a Restrictions f () must have nonzero slope Domain of f () : nonnegative values No restrictions Inverses of Simple Quadratic and Cubic Functions 57

5 EXPLAIN Finding the Inverse of a Quadratic Model Eplain Finding the Inverse of a Quadratic Model In man instances, quadratic functions are used to model real-world applications. It is often useful to find and interpret the inverse of a quadratic model. Note that when working with real-world applications, it is more useful to use the notation () for the inverse of () instead of the notation (). Eample Find the inverse of each of the quadratic functions. Use the inverse to solve the application. QUESTIONING STRATEGIES How does the inverse of a function relate to the original function in a real-world application? It tells ou how to find the value of what was the independent variable in the original function, given the value of what was the dependent variable. The function d (t) = 1 t gives the distance d in feet that a dropped object falls in t seconds. Write the inverse function t (d) to find the time t in seconds it takes for an object to fall a distance of d feet. Then estimate how long it will take a penn dropped into a well to fall feet. The original function d (t) = 1 t is a quadratic function with a domain restricted to t 0. Find the inverse function. Write d (t) as d. d = 1 t Divide both sides b 1. Use the definition of positive square root. Write t as t (d). The inverse function is t (d) = d_ for d 0. 1 d_ 1 = t d_ 1 = t d_ 1 = t (d) Use the inverse function to estimate how long it will take a penn dropped into a well to fall CreativeNature.nl/Shutterstock Image Credits: CreativeNature.nl/Shutterstock feet. Substitute d = into the inverse function. Write the function. t (d) = d_ 1 Substitute for d. t () = _ 1 Simplif. t () = _ Use a calculator to estimate. t () 1.7 So, it will take about 1.7 seconds for a penn to fall feet into the well. The function E (v) = v gives the kinetic energ E in Joules of an -kg object that is travelling at a velocit of v meters per second. Write and graph the inverse function v (E) to find the velocit v in meters per second required for an -kg object to have a kinetic energ of E Joules. Then estimate the velocit required for an -kg object to have a kinetic energ of 0 Joules. The original function E (v) = v is a to v 0. quadratic function with a domain restricted Module Lesson Lesson 11.1

6 Find the inverse function. Write E (v) as E. E = v Divide both sides b. E_ = v Use the definition of positive square root. E_ = v Write v as v (E). v (E) = E_ E_ The inverse function is v (E) = for E 0. Use the inverse function to estimate the velocit required for an -kg object to have a kinetic energ of 0 Joules. INTEGRATE TECHNOLOGY A graphing calculator can be used to verif that functions are inverses of each other. Students can graph the two functions and the function f () =, and check that the graphs of their functions are reflections of each other across the graph of f () =. Substitute E = 0 into the inverse function. Write the function. v (E) = Substitute 0 for E. Simplif. Use a calculator to estimate. E_ v = _ 0 v (0) = 15 0 v (0).9 So, an -kg object with kinetic energ of 0 Joules is traveling at a velocit of per second..9 meters Your Turn Find the inverse of the quadratic function. Use the inverse to solve the application. 5. The function A (r) = π r gives the area of a circular object with respect to its radius r. Write the inverse function r (A) to find the radius r required for area of A. Then estimate the radius of a circular object that has an area of 0 c m. The original function has a domain restricted to r 0. A = π r A_ π = r A_ π = r A_ π = r (A) r (A) = A_ π r (0) = _ 0 r (0). π So, a circular object with an area of 0 cm will have a radius of about. cm. Module Lesson 1 Inverses of Simple Quadratic and Cubic Functions 57

7 EXPLAIN Finding and Graphing the Inverse of a Simple Cubic Function QUESTIONING STRATEGIES Wh is there no need to restrict the domain of a simple cubic function before finding its inverse? Because it is a one-to-one function, its inverse is also a function. How does the domain of a cube root function compare with the domain of a square root function? Wh? The domain of a cube root function is all real numbers. The domain of a square root function is the values of that make the epression inside the square root greater than or equal to zero. The square root of a negative number is not a real number, but the cube root of a negative number is a real number. Eplain Finding and Graphing the Inverse of a Simple Cubic Function Note that the function ƒ () = is a one-to-one function, so its domain does not need to be restricted in order to find its inverse function. The inverse of ƒ () = is ƒ () =. The inverse of a cubic function is a cube root function, which is a function whose rule involves. The parent cube root function is g () =. Eample ƒ () = 0. 5 Find the inverse of each cubic function. Confirm the inverse relationship using composition. Graph the function and its inverse. Find each inverse. Graph the function and its inverse. Replace ƒ () with. = 0.5 Multipl both sides b. = Use the definition of cube root. Switch and to write the inverse. Replace with ƒ (). Confirm the inverse relationship using composition. = = = ƒ () ƒ (ƒ()) = ƒ (0.5 ) ƒ (ƒ ()) = ƒ ( ) (0.5 ) = 0.5 ( ) = = = = = 0.5() Since ƒ (ƒ () ) =, and ƒ (ƒ () ) =, it has been confirmed that ƒ () = of ƒ () = 0.5. is the inverse function Graph ƒ () b graphing ƒ () and reflecting the graph over the line = Module Lesson Lesson 11.1

8 B ƒ () = - 9 Find the inverse. Replace ƒ () with. = - 9 Add 9 to both sides. + 9 = Use the definition of cube root. Switch and to write the inverse. Replace with ƒ (). + 9 Confirm the inverse relationship using composition. = ƒ (ƒ()) = ƒ ( ) ƒ (ƒ ()) = ƒ ( ) = ( - 9) + 9 = + 9 = + 9 = f () ( + 9 ) - 9 INTEGRATE MATHEMATICAL PROCESSES Focus on Technolog The table feature on a graphing calculator can be used to verif that two functions are inverses of each other. Students can enter both functions, and use the table to verif that if (, ) belongs to Y1, then (, ) belongs to Y. = = = = Since ƒ (ƒ () ) = and ƒ (ƒ () ) =, it has been confirmed that ƒ () = is the inverse function of ƒ () = - 9. Graph ƒ () b graphing ƒ () and reflecting the graph over the line = Module Lesson 1 Inverses of Simple Quadratic and Cubic Functions 57

9 Your Turn Find each inverse. Graph the function and its inverse.. ƒ () = Find the inverse. Replace f () with. = Divide both sides b. Use the definition of cube root. Switch and to write the inverse. Replace with f (). _ = _ = _ = _ = f () Confirm the inverse relationship using composition. f (f() ) = f ( ) f ( f ()) = f ( _ ) = _ = ( _ ) = = ( ) = = Since f (f () ) = and f ( f ()) =, it has been confirmed that f () = _ function of f () =. Graph f () b graphing f () and reflecting the graph over the line = is the inverse 7. ƒ () = Find the inverse. Replace f () with. = Divide both sides b. Use the definition of cube root. Switch and to write the inverse. Replace with f (). _ = _ = _ = _ = f () Confirm the inverse relationship using composition. f (f()) = f ( ) f ( f ()) = f ( _ ) = _ = ( _ ) = = ( _ ) = = Since f (f () ) = and f ( f ()) =, it has been confirmed that f () = _ function of f () =. Graph f () b graphing f () and reflecting the graph over the line = is the inverse Module Lesson Lesson 11.1

10 Eplain Finding the Inverse of a Cubic Model In man instances, cubic functions are used to model real-world applications. It is often useful to find and interpret the inverse of cubic models. As with quadratic real-world applications, it is more useful to use the notation () for the inverse of () instead of the notation (). Eample Find the inverse of each of the following cubic functions. The function m (L) = L gives the mass m in kilograms of a red snapper of length L centimeters. Find the inverse function L (m) to find the length L in centimeters of a red snapper that has a mass of m kilograms. The original function m (L) = L is a cubic function. Find the inverse function. Write m (L) as m. m = L Multipl both sides b 100, ,000m = L Use the definition of cube root. Write L as L (m). The inverse function is L(m) = 100,000m = L 100,000m = L (m) 100, 000m. The function A (r) = _ π r gives the surface area A of a sphere with radius r. Find the inverse function r (A) to find the radius r of a sphere with surface area A. The original function A (r) = _ π r is a cubic function. Find the inverse function. Write A (r) as A. A = _ π r Divide both sides b _ π. π A = r Use the definition of cube root. Write r as r (A). _ The inverse function is r(a) = π A. _ _ π A _ = r π A = r (A) Image Credits: Vladimir Melnik/Shutterstock EXPLAIN Finding the Inverse of a Cubic Model QUESTIONING STRATEGIES How do ou determine the domain and range of the inverse of the function in a real-world application? The domain is the range of the original function, and the range is the domain of the original function. Domains ma need to be restricted to values that are reasonable values of the independent variable in the given contet. INTEGRATE MATHEMATICAL PROCESSES Focus on Communication To ensure that students understand the relationship between the function and its inverse in this contet, ask them to describe real-world questions that could more easil be answered using the original function, and those that could more easil be answered using the inverse function. Module Lesson 1 Inverses of Simple Quadratic and Cubic Functions 50

11 ELABORATE INTEGRATE MATHEMATICAL PROCESSES Focus on Critical Thinking Ask students to consider whether the end behavior of the graph of a function can be used to determine the need to restrict the domain in order for the inverse to be a function. Lead them to recognize that when the end behavior is the same as approaches either or - (as is the case with a quadratic function), it is an indicator. However, when this is not the case, it is not possible to tell whether the function is one-to-one. Have students consider, for eample, the graph of a cubic function that has a local maimum and a local minimum. This function is not one-to-one, but other cubic functions, such as f () =, are. Your Turn. The function m (r) = _ π r gives the mass in grams of a spherical lead ball with a radius of r centimeters. Find the inverse _ function r (m) to find the radius r of a lead sphere with mass m. m = π r _ π m = r _ π m = r _ π m = r (m) Elaborate 9. What is the general form of the inverse function for the function ƒ () = a? State an restrictions on the domains. The inverse of f () = a is f () = _ a, where the domain of both functions must be restricted to { 0}. 10. What is the general form of the inverse function for the function ƒ () = a? State an restrictions on the domains.. There are no restrictions on the domains. The inverse of f () = a is f () = _ a SUMMARIZE THE LESSON How do ou find the function that is the inverse of a quadratic function? What tpe of function is the inverse? Substitute for f (), solve for, and then switch and. The result is a square root function. You must restrict the domain of the original function so that the inverse will be a function. 11. Essential Question Check-In Wh must the domain be restricted when finding the inverse of a quadratic function, but not when finding the inverse of a cubic function? The inverse function of a quadratic function is a square root function, which is not defined for negative values of, whereas the inverse function of a cubic function is a cube root function, which is defined for all real number values of. Module Lesson 1 51 Lesson 11.1 LANGUAGE SUPPORT Communicate Math Have students work in pairs. Ask them to discuss how to fill in a chart like the one below and, when the reach agreement about an entr, to take turns adding it to the chart. Tpe of Function Eample equation Graph of function Quadratic function Inverse: Cubic function Inverse:

12 Evaluate: Homework and Practice EVALUATE Restrict the domain of the quadratic function and find its inverse. Confirm the inverse relationship using composition. Graph the function and its inverse. Online Homework Hints and Help Etra Practice 1. ƒ () = 0. Restrict the domain to { 0}. = 0. f (f()) 5 = 5 = 5 = 5 = f (). ƒ () = f (f ()) = f (0. ) = f ( 5 ) 5 (0. ) = 0. ( = = = 0.(5) 5 ) = for 0 = for 0 Since f (f()) = for 0 and f (f ()) = for 0, it has been confirmed that f () = 5 for 0 is the inverse function of f () = 0. for 0. Restrict the domain to { 0}. f (f()) = _ = _ = _ = _ = f () f (f ()) ) = f ( ) = f ( _ = _ = ( _ = ( _ ) = ) = for 0 = for 0 Since f (f()) = for 0 and f (f ()) = for 0, it has been for 0 is the inverse function of f () = for 0.. ƒ () = + 10 Restrict the domain to { 0}. confirmed that f () = _ = = - 10 = - 10 = - 10 = f () f (f () ) = f ( + 10) = ( + 10) - 10 = = for 0 f (f ()) = f ( 0 ) = ( 0 ) + 10 = = for 0 Since f (f()) = for 0 and f (f ()) = for 10, it has been confirmed that f () = - 10 for 10 is the inverse function of f () = + 10 for ASSIGNMENT GUIDE Concepts and Skills Eplore Finding the Inverse of a Man-to- One Function Eample 1 Finding and Graphing the Inverse of a Simple Quadratic Function Eample Finding the Inverse of a Quadratic Model Eample Finding and Graphing the Inverse of a Simple Cubic Function Eample Finding the Inverse of a Cubic Model QUESTIONING STRATEGIES Practice Eercises 1 Eercises 7 Eercises 9 1 Eercises 1 15 Wh is it important to know whether a function is one-to-one when finding its inverse? Because if it is not one-to-one, ou need to restrict the domain for the inverse to be a function. Module 11 5 Lesson 1 Eercise Depth of Knowledge (D.O.K.) Mathematical Processes 1 Skills/Concepts 1.G Eplain and justif arguments 7 Strategic Thinking 1.A Everda life 9 10 Skills/Concepts 1.D Multiple representations 11 1 Skills/Concepts 1.F Analze relationships 1 15 Strategic Thinking 1.A Everda life 1 17 Strategic Thinking 1.F Analze relationships 1 Strategic Thinking 1.A Everda life Inverses of Simple Quadratic and Cubic Functions 5

13 AVOID COMMON ERRORS Students ma, in error, believe that if the graph of a relation passes the horizontal line test, then the relation is one-to-one. Help them to see that the horizontal line test can be used onl to test whether a function is one-to-one. In other words, the graph must first pass the vertical line test. Restrict the domain of the quadratic function and find its inverse. Confirm the inverse relationship using composition.. ƒ () = 15 Restrict the domain to { 0}. f (f()) = f ( 15 ) = 15 _ 15 = _ 15 = _ 15 = _ 15 = f () Since f (f () ) = for 0 and f ( f ()) = for 0, it has been confirmed that f () = _ 15 for 0 is the inverse function of f () = 15 for ƒ () = - _ Restrict the domain to 0. = - _ + _ = _ + = _ + = _ + = f () Since f (f()) = for 0 and f ( f ()) _ = for - f () = _ + for - is the inverse function of f () = -. ƒ () = 0.7 Restrict the domain to { 0}. = 0.7 _ 10 7 = _ 10 7 = _ 10 7 = _ 10 7 = f () _ = = = for 0 f (f () ) = f ( - _ ) = = ( _ - ) _ + = for 0 f (f () ) = f ( 7_ 10 ) = _ 10 7 ( 7_ 10 ) = = for 0 f ( f ()) = f ( _ 15 ) = 15 ( _ 15 ) = 15 ( _ 15 ) = for 0 f ( f ()) = f ( + _ ) = ( _ + ) = + -, it has been confirmed that for 0. - _ = for - _ f ( f ()) = f ( _ 10 7 ) = 0.7 ( _ 10 7 = 7 ) 10 ( 10 7 ) = for 0 Since f (f () ) = for 0 and f ( f ()) = for 0, it has been confirmed that _ f () = 10 7 for 0 is the inverse function of f () = 0.7 for 0. Module 11 5 Lesson 1 5 Lesson 11.1

14 1 7. The function d (s) = 1.9 s models the average depth d in feet of the water over which a tsunami travels, where s is the speed in miles per hour. Write the inverse function s (d) to find the speed required for a depth of d feet. Then estimate the speed of a tsunami over water with an average depth of 1500 feet. d (s) = d = 1.9d = s 1_ 1.9 s 1_ 1.9 s 1.9d = s 1.9d = s (d) s (d) = 1.9d s (1500) = 1.9 (1500) s (1500) 150 So, the speed of a tsunami over water with an average depth of 1500 feet is about 150 mi/h. INTEGRATE MATHEMATICAL PROCESSES Focus on Modeling Discuss with students how a function and its inverse can both model a given real-world situation, even if one is a power function and the other is a radical function.. The period of a pendulum is the time it takes the pendulum to complete one back-and-forth swing. The function (T) = 9. T ( π ) gives the length in meters for a pendulum that has a period of T seconds. Write the inverse function to find the period of a pendulum that has length. Then find the period of a pendulum with a length of 5 meters. (T) = 9. ( T_ π ) = 9. ( T_ π ) _ 9. = ( T_ π ) _ T_ 9. = π π _ 9. = T π _ 9. = T () T () = π _ 9. T (5) = π 5_ T (5).5 9. The period of a pendulum with a length of 5 meters is about.5 seconds. Image Credits: (t) Nejron Photo/Shutterstock; (b) James Stevenson/Science Photo Librar Module 11 5 Lesson 1 Inverses of Simple Quadratic and Cubic Functions 5

15 MULTIPLE REPRESENTATIONS Discuss with students how smbolic representations, graphs, and tables of values can all be used to analze the relationship between a quadratic or cubic function and its inverse. Help students to make connections among the various representations. Different learners ma find that one tpe of representation is more useful than others in aiding their understanding. Find the inverse of each cubic function. Confirm the inverse relationship using composition. Graph the function and its inverse. 9. ƒ () = 0.5 = 0.5 = = = = f () Since f (f () ) = and f ( f ()) =, it has been confirmed that f () = is the inverse function of f () = 0.5. f (f () ) = f (0.5 ) = (0.5 ) = = f ( f ()) = f ( ) = 0.5 ( ) = 0.5() = ƒ () = = = - 1 = - _ 1 = - _ 1 = f () f (f () ) = f ( ) = _ = = f ( f ()) = f ( _ - 1 ) = ( = ( _ - 1 ) = Since f (f () ) = and f ( f ()) =, it has been confirmed that f () = - is the 1 inverse function of f () =. _ _ - 1 ) Find the inverse of the cubic function. Confirm the inverse relationship using composition. 11. ƒ () = - _ 5 = - 5_ + 5_ = 5_ + = 5_ + = 5_ + = f () Since f (f () ) = and f ( f ()) =, it has been confirmed that f () = inverse function of f () = - 5_. f (f () ) = f ( - 5_ = ) ( 5_ - ) 5_ + = = f ( f ()) = f ( + 5_ ) = ( 5_ + ) 5_ 5_ = + - = - 5_ 5_ + is the Module Lesson 1 55 Lesson 11.1

16 1. ƒ () = + 9 = = - 9 = - 9 = - 9 = f () Since f (f () ) = and f ( f ()) =, f () = - 9 is the inverse function of f () = The function m (r) = 1 r models the mass in grams of a spherical zinc ball as a function of the ball s radius in centimeters. Write the inverse model to represent the radius r in cm of a spherical zinc ball as a function of the ball s mass m in g. m (r) = 1 r m = 1 r m_ 1 = r m_ 1 = r m_ 1 = r (m) f (f () ) = f ( + 9) = ( + 9) - 9 = = f ( f ()) = f ( -9 ) = ( -9 ) + 9 = = INTEGRATE TECHNOLOGY Encourage students to use both the graphing feature and the table feature on a graphing calculator to help solve problems and to check their work. 1. The function m (r) = 1 r models the mass in grams of a spherical titanium ball as a function of the ball s radius in centimeters. Write the inverse model to represent the radius r in centimeters of a spherical titanium ball as a function of the ball s mass m in grams. m (r) = 1 r m = 1 r m_ 1 = r m_ 1 = r m_ 1 = r (m) 15. The weight w in pounds that a shelf can support can be modeled b w (d) =.9 d where d is the distance, in inches, between the supports for the shelf. Write the inverse model to represent the distance d in inches between the supports of a shelf as a function of the weight w in pounds that the shelf can support. w (d) =.9 d w =.9 d w_.9 = d w_.9 = d w_.9 = d (w) Image Credits: oorka/ Shutterstock Module 11 5 Lesson 1 Inverses of Simple Quadratic and Cubic Functions 5

17 PEER-TO-PEER DISCUSSION Ask students to discuss with a partner how restricting the domain of f () = to 0, instead of to 0, affects the domain and range of the inverse of the function. The domain of the inverse function will be the same as if the restriction were 0, but the range will be f () 0, instead of f () 0. The inverse function would be f () = -. JOURNAL Have students describe the functions that are inverses of quadratic functions and cubic functions. Have them eplain wh it is sometimes necessar to restrict the domain of a function when finding its inverse. H.O.T. Focus on Higher Order Thinking 1. Eplain the Error A student was asked to find the inverse of the function ƒ () = ( ) + 9. What did the student do wrong? Find the correct inverse. ƒ () = ( _ The student multiplied both sides b before taking the cube root of both sides. The student should have taken the cube root ) + 9 = ( _ ) + 9 ) - 9 = ( _ - 1 = - 1 = = - 1 ƒ () = - 1 of both sides first, and then multiplied b. f () = ( _ ) + 9 = ( _ ) = ( _ ) _ - 9 = - 9 = = - 9 f () = Make a Conjecture The function ƒ () = must have its domain restricted to have its inverse be a function. The function ƒ () = does not need to have its domain restricted to have its inverse be a function. Make a conjecture about which power functions need to have their domains restricted to have their inverses be functions and which do not. A power function whose power is even will have to have its domain restricted. A power function whose power is odd does not have to have its domain restricted. c Multi-Step A framing store uses the function ( 0.5 ) = a to determine the total area of a piece of glass with respect to the cost before installation of the glass. Write the inverse function for the cost c in dollars of glass for a picture with an area of a in square inches. Then write a new function to represent the total cost C the store charges if it costs $.00 for installation. Use the total cost function to estimate the cost if the area of the glass is 19 cm². c - 0. (_ 0.5 ) = a C = c + _ c - 0. = C = 0.5 a a 0.5 c - 0. = C = a c = 0.5 C = 1.1 a + 0. The total cost with installation would be $1.1. Module Lesson 1 57 Lesson 11.1

18 Lesson Performance Task CONNECT VOCABULARY One method used to irrigate crops is the center-pivot irrigation sstem. In this method, sprinklers rotate in a circle to water crops. The challenge for the farmer is to determine where to place the pivot in order to water the desired number of acres. The farmer knows the area but needs to find the radius of the circle necessar to define that area. How can the farmer determine this from the formula for the area of a circle A = π r? Find the formula the farmer could use to determine the radius necessar to irrigate a given number of acres, A. (Hint: One acre is,50 square feet.) What would be the radius necessar for the sprinklers to irrigate an area of 1 acres? Area = πr r Some students ma not be familiar with the term pivot. Eplain to students that a pivot is a point from which something rotates. Draw a circle, and mark the center of the circle. Eplain to students that a rotating sprinkler defines a circle, and the pivot point, where the sprinkler is located, is at the center of the circle. Use the area of a circle. A = π r,50a = π r _,50A π = _ π r π _,50A π = r _,50A π = r 11 A r B substituting in the number of acres for A, ou get the radius necessar to define a circle with the same area as that number of acres. The radius necessar to irrigate an area of 1 acres would be: r 11 A r 11 1 r 11 ft 1_ This would be a little over of a mile. QUESTIONING STRATEGIES Wh is a quadratic function used in this problem? Area is a two-dimensional quantit, and when both dimensions are the same, the area is proportional to the square of the dimension. For eample, the area of a circle is proportional to the radius squared. What are the domain and range of the inverse function r = 11 A, and how are the restricted b the real-world situation? The domain is A 0, and the range is r 0. Both quantities must be greater than or equal to zero because length and area cannot be negative. Module 11 5 Lesson 1 EXTENSION ACTIVITY Eplain to students that the farmer is tring to conserve water and is limited to million gallons of water for each irrigation. If the crops need inches of water per irrigation to be productive, have students calculate the maimum radius of the circular area the farmer can irrigate. (Hint: 1 gallon is approimatel 0.1 cubic feet.) The volume of water is given b V = πr d, where d is the depth of water. Solving for the inverse function gives r = V_ πd =,000,000 (0.1) _ π ( 1 ) = feet. Scoring Rubric points: Student correctl solves the problem and eplains his/her reasoning. 1 point: Student shows good understanding of the problem but does not full solve or eplain his/her reasoning. 0 points: Student does not demonstrate understanding of the problem. Inverses of Simple Quadratic and Cubic Functions 5