10.1 Inverses of Simple Quadratic and Cubic Functions

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1 COMMON CORE Locker LESSON 0. Inverses of Simple Quadratic and Cubic Functions Name Class Date 0. 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? Resource Locker Common Core Math Standards The student is epected to: COMMON CORE F-BF.B.a Solve an equation of the form f() c for a simple function f that has an inverse and write an epression for the inverse. Also F-BF.B.b(+), F-BF.B.c(+), F-BF.B.d(+) Mathematical Practices COMMON CORE MP. Reasoning Language Objective 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. Eplore Finding the Inverse of a Man-to-One Function The function ƒ () is defined b the following ordered pairs: (-, ), (, ), (0, 0), (, ), and (, ). A Find the inverse function of ƒ (), ƒ (), b reversing the coordinates in the ordered pairs. (, -), (, ), (0, 0), (, ), (, ) B Is the inverse also a function? Eplain. No; the -values and are both paired with more than one -value. C 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 D With the restricted domain of ƒ (), what ordered pairs define the inverse function ƒ ()? (0, 0), (, ), (, ) Reflect. Discussion Look again at the ordered pairs that define ƒ (). Without reversing the order of 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 coordinates 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. 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. Module 0 79 Lesson Name Class Date 0. 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? F-BF.B.a Solve an equation of the form f() c for a simple function f that has an inverse and write an epression for the inverse. Also F-BF.B.b(+), F-BF.B.c(+), F-BF.B.d(+) Eplore Finding the Inverse of a Man-to-One Function The function ƒ () is defined b the following ordered pairs: (-, ), (, ), (0, 0), (, ), and (, ). Find the inverse function of ƒ (), ƒ (), b reversing the coordinates in the ordered pairs. (, -), (, ), (0, 0), (, ), (, ) 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), (, ), (, ). Discussion Look again at the ordered pairs that define ƒ (). Without reversing the order of the coordinates, how could ou have known that the inverse of ƒ () would not be a function?. How will restricting the domain of ƒ () affect the range of its inverse? Resource When two or more ordered pairs have the same -value but unique -values, the inverse of that function will have coordinates 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. 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 0 79 Lesson HARDCOVER PAGES 5 5 Turn to these pages to find this lesson in the hardcover student edition. 79 Lesson 0.

2 Eplain 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 ƒ () 0.5 Restrict the domain of each quadratic function and find its inverse. 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 (). _ _ 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. 0 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 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 0 80 Lesson PROFESSIONAL DEVELOPMENT Math Background The domain of an even function f () n where n,, 6, 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,, 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 80

3 INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP. 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 Find the inverse. Restrict the domain. 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 () ƒ (ƒ()) ƒ ( ) ƒ ( ƒ ()) f ( ) f () ( ( - 7-7) ) for f () -7 ƒ () + 7 for -7 0 is the inverse function of ƒ () - 7 for. Since ƒ (ƒ () ) for and ƒ ( ) for, it has been confirmed that Graph ƒ () b graphing ƒ () over the restricted domain and reflecting the graph over the line Module 0 8 Lesson 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 (). 8 Lesson 0.

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 (). 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. The function d (t) 6 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 8 feet. The original function d (t) 6 t is a quadratic function with a domain restricted to t 0. Find the inverse function. f () 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. 0 Houghton Mifflin Harcourt Publishing Compan Image Credit: CreativeNature.nl/Shutterstock 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. EXPLAIN Finding the Inverse of a Quadratic Model 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. Write d (t) as d. d 6 t Divide both sides b 6. Use the definition of positive square root. d_ 6 t d_ 6 t Module 0 8 Lesson 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 8

5 Write t as t (d). The inverse function is t (d) d_ for d 0. 6 d_ 6 t (d) Use the inverse function to estimate how long it will take a penn dropped into a well to fall 8 feet. Substitute d 8 into the inverse function. Write the function. t (d) d_ 6 Substitute 8 for d. t (8) _ 8 6 Simplif. t (8) _ Use a calculator to estimate. t (8).7 So, it will take about.7 seconds for a penn to fall 8 feet into the well. B The function E (v) v gives the kinetic energ E in Joules of an 8-kg object that is traveling 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 8-kg object to have a kinetic energ of E Joules. Then estimate the velocit required for an 8-kg object to have a kinetic energ of 60 Joules. The original function E (v) v is a quadratic function with a domain restricted to v 0. Find the inverse function. Write E (v) as E. E v Image Credits: CreativeNature.nl/Shutterstock 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 8-kg object to have a kinetic energ of 60 Joules. Substitute E 60 into the inverse function. Write the function. v (E) Substitute 60 for E. Simplif. Use a calculator to estimate. E_ v _ 60 v (60) 5 60 v (60).9 So, an 8-kg object with kinetic energ of 60 Joules is traveling at a velocit of per second..9 meters Module 0 8 Lesson 8 Lesson 0.

6 Your Turn Find the inverse of the quadratic function. Use the inverse to solve the application.. 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. 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 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).6 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 (). ƒ (ƒ()) ƒ (0.5 ) ƒ (ƒ ()) ƒ ( ) π So, a circular object with an area of 0 cm will have a radius of about.6 cm. (0.5 ) 0.5 ( ) 0.5 () ƒ () 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 (). 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. Module 0 8 Lesson Inverses of Simple Quadratic and Cubic Functions 8

7 Since ƒ (ƒ () ) and ƒ (ƒ () ), it has been confirmed that ƒ () is the inverse function of ƒ () 0.5. Graph ƒ () b graphing ƒ () and reflecting ƒ () over the line 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 () f () ƒ (ƒ()) ƒ ( ) ƒ (ƒ ()) ƒ ( + 9 ) - 9 ( - 9) + 9 Since ƒ (ƒ () ) and ƒ (ƒ () ), it has been confirmed that ƒ () is the inverse function of ƒ () - 9. ( + 9 ) Module 0 85 Lesson 85 Lesson 0.

8 Graph ƒ () b graphing ƒ () and reflecting ƒ () over the line INTEGRATE MATHEMATICAL PRACTICES Focus on Technolog MP.5 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 Y, then (, ) belongs to Y. Your Turn Find each inverse. Graph the function and its inverse. 5. 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 (). 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 (). _ _ _ _ f () 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 f () over the line. ) - - is the inverse 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. Module 0 86 Lesson Inverses of Simple Quadratic and Cubic Functions 86

9 INTEGRATE MATHEMATICAL PRACTICES Focus on Communication MP. 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. 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 00, ,000m L Use the definition of cube root. Write L as L (m). The inverse function is L(m) 00,000m L 00,000m L (m) 00, 000m. Image Credits: Vladimir Melnik/Shutterstock 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. Your Turn _ 6. 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 _ πr _ π m r π m r π m r (m) _ _ π A _ r π A r (A) Module 0 87 Lesson 87 Lesson 0.

10 Elaborate 7. What is the general form of the inverse function for the function ƒ () a? State an restrictions on the domains., where the domain of both functions must be The inverse of f () a is f () _ a restricted to { 0}. 8. 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 () 9. 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 nonnegative values of, whereas the inverse function of a cubic function is a cube root function, which is defined for all real number values of. Evaluate: Homework and Practice Restrict the domain of the quadratic function and find its inverse. Graph the function and its inverse. _ a. There are no restrictions on the domains.. ƒ () 0. 8 Restrict the domain to { 0}. 0. f (f()) f (0. ) f () f (f ()) f ( 5 ) 0. ( 5 ) 0. (5) for 0 5 (0. ) for 0 Since f (f()) and f ( f ()) for 0, it has been confirmed that f () 5 for 0 is the inverse function of f () 0. for 0. 0 Online Homework Hints and Help Etra Practice 6 8 ELABORATE INTEGRATE MATHEMATICAL PRACTICES Focus on Critical Thinking MP. 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. 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. Module 0 88 Lesson 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: Inverses of Simple Quadratic and Cubic Functions 88

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