{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 128 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 128 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 128 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 128 128 1 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 128 128 1 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 267 "" 0 1 128 128 128 1 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "N ormal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 0 "" 0 "" {TEXT 263 49 "High School Modules > Algebra by Gregory A. Moore" }}{PARA 3 "" 0 "" {TEXT -1 4 " " }{TEXT 262 20 " One-to-One Functions" }}{PARA 0 "" 0 "" {TEXT -1 67 "\nThe concept of \+ one-to-one functions and the horizontal line test.\n" }}{PARA 0 "" 0 " " {TEXT 264 1 "[" }{TEXT 265 10 "Directions" }{TEXT 266 142 " : Execut e the Code Resource section first. Although there will be no output im mediately, these definitions are used later in this worksheet.]" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 7 "0. Code" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "restart; with(plots ): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 161 "Domain := proc( S ET )\n local domain, k;\n domain := \{\};\n for k from 1 to nops (SET) do\n domain := domain union \{SET[k][1]\};\n od:\n domai n;\nend proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 160 "Range := proc( SET )\n local range, k;\n range := \{\};\n for k from 1 \+ to nops(SET) do\n range := range union \{SET[k][2]\};\n od:\n \+ range;\nend proc: \n " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 38 "1. One-to-One Functions - S et Approach" }}{PARA 0 "" 0 "" {TEXT -1 225 "\nWhen we consider abstra ct functions as sets of pairings, it is perfectly possible that two di fferent values of the domain may share the value of the range. For exa mple, look at this set. Various fruits share the same colors." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "fruity_colors := \{ [lemon, yellow], [grape, purple], \n [banana, yellow], [app le,red],[cherry,red] \};" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "nature_colors := \{ [grass, green], [frog, green],[fox,red], \n \+ [mouse, brown], [moose,brown] \};" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "class_hair_colors := \{ [Alan, blond], [Be atriz, brunette],[Carlos,blond], \n [Darby, black_ha ir], [Eiko,black_hair] \};" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "monthly_temps := \{ [January, cold], [April, temperate],[June,war m], \n [Augst, warm], [December,cold] \};" }}}{PARA 0 "" 0 "" {TEXT -1 155 "\nHowever, in certain special cases, each doma in value may have a unique range value. We call such functions \"one t o one\" functions. Here are some examples." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 207 "class_addresses := \{ [Alan, `123 Maple Ave` ], [B eatriz, `456 ElmStreet` ],\n [Carlos, `1201 Main P lace`], \n [Darby, `3000 Old Mill Road` ], [Eiko, `4 201 Beach Blvd` ] \};" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "c lass_ID_numbers := \{ [Alan, 12345 ], [Beatriz, 80012 ],\n \+ [Carlos, 11258], \n [Darby, 01248], [Eiko, \+ 52526 ] \};" }}}{PARA 0 "" 0 "" {TEXT -1 153 "\nNotice that one-to-one functions of finite sets always have the same size of domain and rang e. That is not necessarily true for non-one-to-one functions" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "Domain(class_addresses); Ran ge(class_addresses); nops(%) = nops(%%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "Domain(class_ID_numbers); Range(class_ID_numbers); no ps(%) = nops(%%); " }}}{PARA 0 "" 0 "" {TEXT -1 29 "\n\nWhich of these number sets " }{TEXT 261 6 "appear" }{TEXT -1 35 " to represent one-t o-one functions?" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "Number_s et := \{ seq([k, 3*k + 2], k = 1..9) \};" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "Number_set := \{ seq([k, 100-abs(k-5)], k = 1..9) \}; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "Number_set := \{ seq([k , 100 + 50*(abs( k+3) - abs(k-3))], k = 1..9) \};" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 63 "Number_set := \{ seq([k, floor( 100*sin(k*Pi /10))], k = 1..9) \};" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "Nu mber_set := \{ seq([k, (k+2)^3 - 5], k = 1..9) \};" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 44 "2. One-to-One Functions - Numerical Approach" } }{PARA 0 "" 0 "" {TEXT -1 531 "\nA function is \"one to one\" if each \+ x value has a unique y value. In other words, no two x values have the same y value. Not all functions are one-to-one. This is a special pro perty only belonging to certain functions. This property is analogous \+ to requiring that all marriages be monogomous - each person can have o nly one spouse. On the other hand, a function which does not have this property would have at least two different x values which have the sa me y value - like a case where two different women have the same husba nd.\n\n" }{TEXT 256 14 "NOT One-to-One" }{TEXT -1 121 "\n\nHere are ex amples of a functions which are not one-to-one. Can you see find two x values which share the same y value?\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "f:= x -> 10 - (x-5)^2:\narray( [[ k,f(k) ] $ k = 0..1 0] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "f:= x -> abs(x-4) \+ - abs(x-8):\narray( [[ k,f(k) ] $ k = 0..10] );" }}}{PARA 0 "" 0 "" {TEXT -1 2 "\n\n" }{TEXT 257 10 "One-to-One" }{TEXT -1 141 "\n\nHere a re some one-to-one functions. While these tables don't prove that they are one-to-one, you won't find any two x's sharing the same y." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "f:= x -> 10 + (x-4)^3:\narra y( [[ k,f(k) ] $ k = 0..10] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "f:= x -> 2^(x-4) :\narray( [[ k,f(k) ] $ k = 0..10] );" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "f:= x -> floor( 100/(x-5.5)) :\narray( [[ k,f(k) ] $ k = 0..10] );" }}}{PARA 0 "" 0 "" {TEXT -1 2 " \n\n" }{TEXT 258 32 "Showing a Function is One-to-One" }{TEXT -1 207 " \n\nThe way to prove that a function is one-to-one is to assume that t wo different x values, a and b, share the same y, then show that these two \"different\" values are actually the same. Here is an example : \n\n" }{TEXT 260 15 " Example" }{TEXT -1 13 " : Show that " } {MPLTEXT 1 0 0 "" }{TEXT -1 7 "f(x) = " }{XPPEDIT 18 0 "100 + (3*x - 5 )^3" "6#,&\"$+\"\"\"\"*$,&*&\"\"$F%%\"xGF%F%\"\"&!\"\"F)F%" }{TEXT -1 16 " is one-to-one.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "f : = x -> 100 + (3*x - 5)^3;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "f(a) = f(b);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "lhs(%) \+ -100 = rhs(%) - 100;" }}}{PARA 0 "" 0 "" {TEXT -1 35 "take the cube ro ot of each side ..." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "3*a-5 = 3*b- 5;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "lhs(%)+5 = rh s(%) + 5;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "lhs(%)/3 = rhs (%)/3;" }}}{PARA 0 "" 0 "" {TEXT -1 2 "\n " }}}{SECT 0 {PARA 4 "" 0 " " {TEXT -1 23 "3. Horizontal Line Test" }}{PARA 0 "" 0 "" {TEXT -1 239 "\nWhen looking at a graph it is fairly easy to see if its the gra ph of a function or not using the vertical line test. In a similar way , it is also possible to determine if the function is one-to-one or no t using the horizontal line test.\n\n" }{TEXT 259 57 "Horizontal Line \+ Test - (tests for a one-to-one functions)" }{TEXT -1 431 "\n\nOne-to-o ne functions are a special type of function. Each y value on the graph is unique to a particular value of x. Thus, if we draw a horizontal l ine, the graph will only cross in at most one point. If lines were to \+ cross in two or more places, that would mean different x's share the s ame y - and this means the function is not one-to-one. Look at each of the graphs below and decide whether the function is one-to-one or not ." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 198 "f := x -> x^3 - 3*x^2 \+ + x -1;\nplots[display]( plot( f(x), x= -2..4, y=-7..7, color = magent a, thickness=2),\n plot( \{seq( [[-7,k/2],[7,k/2]], k = -14..1 4)\}, x= -2..4, y=-7..7, color =green));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 206 "f := x -> ((x+1)^2 - (x-1)^3 - 10)/3;\nplots[display ]( plot( f(x), x= -2..4, y=-7..7, color = magenta, thickness=2),\n \+ plot( \{seq( [[-7,k/2],[7,k/2]], k = -14..14)\}, x= -2..4, y=-7.. 7, color =green));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 223 "f := x -> ((x)^2 + 4*sin(3*x)*(1+sin(x)) - 5)/2;\nplots[display]( plot( f( x), x= -2..2*Pi, y=-7..17, color = magenta, thickness=2),\n pl ot( \{seq( [[-7,k/2],[17,k/2]], k = -14..34)\}, x= -2..4, y=-7..17, co lor =green));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart: w ith(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 188 "f := x -> . 1^x - 2;\nplots[display]( plot( f(x), x= -3..2, y=-2..6, color = magen ta, thickness=2),\n plot( \{seq( [[-7,k/4],[17,k/4]], k = -8.. 24)\}, x= -3..2, y=-2..6, color =green));" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 197 "f := x -> (5^x - 3^x - 2^x);\nplots[display]( plot ( f(x), x= -3..2, y=-2..6, color = magenta, thickness=2),\n pl ot( \{seq( [[-7,k/4],[17,k/4]], k = -8..24)\}, x= -3..2, y=-2..6, colo r =green));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 199 "f := x -> 5 + 12*cos(x);\nplots[display]( plot( f(x), x= -2..2*Pi, y=-7..17, colo r = magenta, thickness=2),\n plot( \{seq( [[-7,k/2],[17,k/2]], k = -14..34)\}, x= -2..4, y=-7..17, color =green));" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT 267 35 "\n \251 2002 Waterloo Maple Inc" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "0 0" 24 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }