Determine Whether Two Functions Are Equivalent. Determine whether the functions in each pair are equivalent by. and g (x) 5 x 2

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1 .1 Functions and Equivalent Algebraic Epressions On September, 1999, the Mars Climate Orbiter crashed on its first da of orbit. Two scientific groups used different measurement sstems (Imperial and metric) for navigational calculations, resulting in a mi-up that is said to have caused the loss of the $15-million (U.S.) orbiter. Even though the National Aeronautics and Space Administration (NASA) requires a sstem of checks within their processes, this error was never detected. This mi-up ma have been caused b people not understanding comple equations. In general, to reduce the likelihood of errors in calculations, mathematicians and engineers simplif equations and epressions before appling them. Eample 1 Determine Whether Two Functions Are Equivalent Determine whether the functions in each pair are equivalent b i) testing three different values of ii) simplifing the epressions on the right sides iii) graphing using graphing technolog a) f () 5 ( 1) ( ) and g () 5 b) f () 5 8 and g () 5 1 Solution a) i) Choose three values of that will make calculations relativel eas. f () ( 1) + ( ) g () 1 f ( 1) ( 1 1) + (( 1) ) ( ) + ( 5) 8 5 g( 1) ( 1) ( 1) (1) f (0) (0 1) + ((0) ) ( 1) + ( ) g(0) (0) f (1) (1 1) + ((1) ) (0) g(1) (1) MHR Functions 11 Chapter

2 Based on these three calculations, the functions appear to be equivalent. However, three eamples do not prove that the functions are equivalent for ever -value. ii) In this pair, g () is alread simplified, so concentrate on f (). f () 5 ( 1) ( ) 5 ( 1) 5 5 Algebraicall, these functions are equivalent. iii) Use a graphing calculator to graph the two equations as Y1 and Y. Change the line displa for Y to a thick line. Cursor left to the slanted line beside Y. Press ENTER to change the line stle. From the ZOOM menu, select 6:ZStandard. The graph of 5 ( 1) ( ) will be drawn first. Then, the graph of 5 will be drawn using a heavier line. You can pause the plot b pressing ENTER. Pressing ENTER again will resume the plot. This seems to ield the same graph. These functions appear to be equivalent. Using a TI-Nspire CAS graphing calculator, ou can graph the functions f () 5 ( 1) ( ) and g () 5 side b side for comparison. Technolog Tip Refer to the Use Technolog feature on pages 86 and 87 to see how to graph multiple functions using a TI-Nspire CAS graphing calculator..1 Functions and Equivalent Algebraic Epressions MHR 79

3 b) i) f () f ( 1) ( 1) ( 1) 8 ( 1) ( 1) _ _ 0 f (0) 0 (0) 8 0 8_ _ 1 f (1) 1 (1) 8 1 9_ 1 1 _ 1 _ g () + + g ( 1) _ + _ g (0) 0 + _ 0 + _ g (1) 1 + _ 1 + Based on these three calculations, the functions appear to be equivalent. However, three eamples do not prove that these functions are equivalent for ever -value. ii) In this pair, g () is alread simplified, so concentrate on f (). To simplif f (), factor the numerator and the denominator. f () ( )( ) 5 ( )() Factor the numerator and the denominator. 5 Divide b the common factor. Algebraicall, it appears as though the two functions are equivalent. However, the effect of dividing b a common factor involving a variable needs to be eamined. Technolog Tip Using a friendl window, such as ZDecimal, makes it easier to see an gaps in the graph of a function. This is because each piel represents one tick mark. iii) Use a graphing calculator to graph the two equations. In this case, there is a slight difference between the graphs. To see the graphs properl, press ZOOM and select :ZDecimal. gap 80 MHR Functions 11 Chapter

4 There appears to be a gap in the first graph. This can be verified further b using the TABLE function on a graphing calculator. Based on the evidence, this pair of functions is equivalent everwhere but at 5. Connections An open circle is used to indicate a gap or a hole in the graph of a function. Polnomial epressions that can be algebraicall simplified to the same epression are equivalent. However, with rational epressions, this ma not be the case. More specificall, since division b zero is not defined, ou must define restrictions on the variable. For eample, the function f () ( )( ) has a factored form of f () 5. Since the denominator is ( )() zero if 5 0 or the simplified function is written as f () 5,,. f() 0 rational epression the quotient of two _ polnomials, p() q(), where q() 0 Eample Determine Restrictions Simplif each epression and determine an restrictions on the variable. a) 10 1 Solution a) ( )( 7) b) ()( 7) 5, 5 7, So, ,. b) ( 5 _ 1)( 5) 10 ( 5)( ) ( 1)( 5) 5 _ ( 5)( ) So, ,, 5_,, 5_,, 5_. Factor the numerator and the denominator. Before reducing, determine restrictions. In this case,. Divide b an common factors. Factor the numerator and the denominator. Divide b an common factors..1 Functions and Equivalent Algebraic Epressions MHR 81

5 Eample Simplif Calculations A square of side length 7 cm is removed from a square of side length. a) Epress the area of the shaded region as a function of. b) Write the area function in factored form. c) Use both forms of the function to calculate the area for -values of 8 cm, 9 cm, 10 cm, 11 cm, and 1 cm. Which form is easier to use? d) What is the domain of the area function? 7 cm Solution a) A shaded 5 A large A small b) A shaded ( 7)( 7) c) A 9 A ( 7)( + 7) 8 A A A A A A (8 7)(8 + 7) 15 A (9 7)(9 + 7) A (10 7)(10 + 7) 51 A (11 7)(11 + 7) 7 A (1 7)(1 + 7) 95 The areas are 15 cm, cm, 51 cm, 7 cm, and 95 cm, respectivel. Both epressions have similar numbers of steps involved. However, it is easier to use mental math with the factored form. d) Since this function represents area, it must be restricted to -values that do not result in negative or zero areas. So, the domain is { R, 7}. 8 MHR Functions 11 Chapter

6 Ke Concepts To determine if two epressions are equivalent, simplif both to see if the are algebraicall the same. Checking several points ma suggest that two epressions are equivalent, but it does not prove that the are. Rational epressions must be checked for restrictions b determining where the denominator is zero. These restrictions must be stated when the epression is simplified. Graphs can suggest whether two functions or epressions are equivalent. Communicate Your Understanding C1 The points (, 5) and (5, 5) both lie on the graphs of the functions 5 10 and 5 0. Eplain wh checking onl a few points is not sufficient to determine whether two epressions are equivalent. C A student submits the following simplification Eplain how ou would show the student that this is incorrect. C Eplain wh the epression 5 does not have an restrictions. A Practise For help with questions 1 to 6, refer to Eamples 1 and.. Refer to question 1. If the functions appear 1. Use Technolog Use a graphing calculator to graph each pair of functions. Do the appear to be equivalent? a) f () 5 5( ) ( ), g () 5 6 to be equivalent, show that the are algebraicall. Otherwise, show that the are not equivalent b substituting a value for.. State the restriction for each function. a) b) f () 5 (8 ) (5 7)(9 1), g () c) f () 5 ( 5) ( 5), g () b) g () f() e) f () 5 ( 5)( 5 ), g () 5 ( 1) ( 5 1) d) f () 5 ( )( )( 5), 0 0 f() Functions and Equivalent Algebraic Epressions MHR 8 Functions 11 CH0.indd 8 6/10/09 :01:8 PM

7 . State the restrictions for each function. a) b) ( 5) f() 5 f() B Connect and Appl 7. Evaluate each epression for -values of, 1, 0,, and 10. Describe an difficulties that occur. a) ( 6)( ) ( 11)( ) _ b) For help with question 8, refer to Eample. 8. A circle of radius cm is removed from a circle of radius r. 6 0 r cm 5. Determine whether g () is the simplified version of f (). If it is, then state the restrictions needed. If not, determine the correct simplified version. a) f () 5 110, g () b) f () 5 16, g () c) f () 5 6 5, g () 5 5 d) f () , g () e) f () 5 1 5, g () 5 1 f) f () , g () Simplif each epression and state all restrictions on. a) 8 _ 1 0 b) ( 7) ( 10) c) 18 d) e) _ f) a) Epress the area of the shaded region as a function of r. b) State the domain and range of the area function. 9. A compan that Reasoning and Proving makes modular Representing furniture has Problem Solving designed a scalable Connecting bo to accommodate several different Communicating sizes of items. The dimensions are given b L 5 0.5, W 5 0.5, and H 5 0.5, where is in metres. a) Epress the volume of the bo as a function of. b) Epress the surface area of the bo as a function of. c) Determine the volume and surface area for -values of 0.75 m, 1 m, and 1.5 m. d) State the domain and range of the volume and surface area functions. Selecting Tools Reflecting 8 MHR Functions 11 Chapter

8 10. Chapter Problem At the traffic safet bureau, Matthew is conducting a stud on the stoplights at a particular intersection. He determines that when there are 18 green lights per hour, then, on average, 1 cars can safel travel through the intersection on each green light. He also finds that if the number of green lights per hour increases b one, then one fewer car can travel through the intersection per light. a) Determine a function to represent the total number of cars that will travel through the intersection for an increase of green lights per hour. b) Matthew models the situation with the function f () Show that our function from part a) is the same. c) How man green lights should there be per hour to maimize the number of cars through the intersection? 11. In the novel The Curious Incident of the Dog in the Night-Time b Mark Reasoning and Proving Representing Problem Solving Haddon, the oung Connecting Reflecting bo, who is the Communicating main character, loves mathematics and is mildl autistic. Throughout the book, he encounters several math problems. One of the problems asks him to prove that a triangle with sides given b 1, 1, and will alwas be a right triangle for > 1. Selecting Tools a) Use the Pthagorean theorem to verif that this statement is true for -values of,, and. b) Based on the three epressions for the sides, which one must represent the hpotenuse? Justif our answer. c) Use the Pthagorean theorem with the epressions for the side lengths to prove that these will alwas be sides of a right triangle for 1. a 1. The function 5 is sometimes a called the witch of Agnesi after Maria Gaetana Agnesi ( ). The equation generates a famil of functions for different values of a R. a) Use Technolog Use graphing technolog to graph this function for a-values of 1,,, and. b) Eplain wh this rational function does not have an restrictions. c) Research the histor of Maria Gaetana Agnesi and find out who else studied this curve before her. Connections Maria Gaetana Agnesi originall referred to this function as versiera, which means to turn. Later, in translation, it was mistakenl confused with avversiere, which means witch or wife of the devil, and thus its current name was born. C Etend 1. What does the graph of _ f () 5 ( 6)( 6) look like? 1 1. Algebraicall determine the domain and range of the area function that represents the shaded region. 10 cm 15. A student wrote the following proof.what mistake did the student make? 16. Math Contest Given the two linear functions and 5 6, what ordered pair lies on the graph of the first line but not on the graph of the second line?.1 Functions and Equivalent Algebraic Epressions MHR 85

9 Use Technolog Tools TI-Nspire CAS graphing calculator Graph Functions Using a TI-Nspire CAS Graphing Calculator 1. Open a new document. Open a page using the Graphs & Geometr application.. In the entr line, ou will see f1()5. Tpe as a sample function. Press. Note that the function is displaed with its equation as a label and the entr line has changed to f()5.. Look at the aes. This is the standard window. To view or change the window settings: Press b. Select :Window, and then select 1:Window Settings. You can change the appearance of the window. Press b. Select :View. There are several options. For eample, if ou select 8:Show Aes End Values, ou can displa the range of each ais. Technolog Tip You can change the name of a function. For eample, to change f1() to g(): In the entr line, press. several times to erase f1. Tpe g, and press. The function will be displaed with the desired name as the label.. Press the up arrow ke once. The function f1() will be displaed in the entr line. You can change the appearance of a line: Press e until the Attributes tool, $, at the left of the entr line, is selected. Press. You can use this tool to adjust the line weight, the line stle, the label stle, and the line continuit. Use the up and down arrow kes to highlight an attribute. Then, use the left and right arrow kes to move through the options for that attribute. Eperiment with the attributes. When ou are finished, press d. 5. You can move a function label. Press e. The entr line will gre out, and the cursor will move to the graphing window. Use the arrow kes to move the cursor over the function label. When ou are in the correct place, the word label will appear, along with a hand smbol. 86 MHR Functions 11 Chapter

10 Press /. The hand will close to grab the label. Use the arrow kes to move the label around the screen. When ou are finished, press d. You can also move the entire graph. Move the cursor to a blank space in the second quadrant. Press /. A hand will appear. Use the arrow kes to move the entire graph around the screen. When ou are finished, press d. Technolog Tip If ou are on the entr line and want to move to the graphing window, press d. If ou are in the graphing window and want to move to the entr line, press e. 6. You can displa a table of values for the function. Press e to return to the entr line. Press the up arrow ke to return to the function f1(). Press b and select :View. From the View menu, select 9:Add Function Table. You can scroll up and down to inspect different values. To adjust the Table Start value and the Table Step value: Press b and select 5:Function Table. From the Function Table menu, select :Edit Function Table Settings. 7. You can split the screen to displa two functions at once. Open a new document. Open a page using the Graphs & Geometr application. Graph the function f1() 5. Press / c to access the Tools menu. From the Tools menu, select 5:Page Laout, and then select :Select Laout. You will see a menu of possible laouts. For eample, to displa two graphs side b side: Select :Laout. A blank window will appear. Press / e to switch windows. Press b and select :Add Graphs & Geometr. Graph the function f() 5. Technolog Tip You can hide the entr line. Press / G. The entr line is hidden. Press / G again to view the entr line. If ou press a ke or make a selection b mistake, ou can undo the operation. Press / Z. This will work several times to step back through a series of operations. Use Technolog: Graph Functions Using a TI-Nspire CAS Graphing Calculator MHR 87