{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 "" -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 0 128 0 1 0 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 0 128 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 128 128 128 1 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -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 262 49 "High School Modules > Algebra by Gregory A. Moore" }}{PARA 3 "" 0 "" {TEXT -1 4 " " }{TEXT 261 24 " The Algebra of Functions" }}{PARA 0 "" 0 "" {TEXT -1 80 "\nCombining f unctions by sums, products, differences, quotients, and compostion.\n " }}{PARA 0 "" 0 "" {TEXT 263 153 "[Directions : Execute the Code Reso urce section first. Although there will be no output immediately, thes e definitions are used later in this worksheet.]" }{TEXT -1 1 "\n" }} {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 0 853 "FuncOps := proc( f, g, a, b )\n local N, M, \+ P, i,j,k, A, p ;\n \n A := array( [seq( [ \n \+ seq( ` `, j = 1..7 ) \n ], i = a..b+1 ) \+ ]);\n \n A[1,1] := `x`; A[1,2] := `f(x)`; A[1,3] := \+ `g(x)`;\n A[1,4] := `f(x)+g(x)`; A[1,5] := `f(x)-g(x)`;\n \+ A[1,6] := `f(x)*g(x)`; A[1,7] := `f(x)/g(x)`;\n \n for i fr om a to b do A[i+1, 1] := i; od;\n for i from a to b do A[i+1, 2] := f(i); od;\n for i from a to b do A[i+1, 3] := g(i); od;\n f or i from a to b do A[i+1, 4] := f(i)+g(i); od;\n for i from a to \+ b do A[i+1, 5] := f(i)-g(i); od;\n for i from a to b do A[i+1, 6] := f(i)*g(i); od;\n for i from a to b do \n if( g(i) = 0) \+ then A[i+1, 7] := `undef`; \n else A[i+1, 7] := f (i)/g(i); fi; od;\n \n print(A);\n\n end proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 455 "FuncComp := proc( f, g, a, b )\n local N, M, P, i,j,k, A, p ;\n \n A := array( [seq( [ \n \+ seq( ` `, j = 1..3 ) \n ], i \+ = a..b+1 ) ]);\n \n A[1,1] := `x`; A[1,2] := `g(x) `; A[1,3] := `f(g(x))`;\n \n \n for i from a to b do A [i+1, 1] := i; od;\n for i from a to b do A[i+1, 2] := g(i); od;\n for i from a to b do A[i+1, 3] := f(g(i)); od;\n \n print( A);\n\n end proc:" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 53 "1. Arithme tic Operations on Functions - Algebraically" }}{PARA 0 "" 0 "" {TEXT -1 277 "\nThe four basic arithmetic operations of adding, subtracting, multiplying, and dividing, can be performed on functions in the same \+ way one would perform these operations on numbers or expressons. Its p retty straight forward. When adding and subtracting you combine like t erms.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 155 "f := x -> 3*x + \+ 2; \ng := x -> 8*x - 7; \n`f(x) + g(x)` = f(x) + g(x);\n`f(x) - g(x)` \+ = f(x) - g(x);\n`f(x) * g(x)` = f(x) * g(x);\n`f(x) / g(x)` = f(x) / g (x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "f := x -> x; \ng : = x -> 1/x; \n`f(x) + g(x)` = f(x) + g(x);\n`f(x) - g(x)` = f(x) - g(x );\n`f(x) * g(x)` = f(x) * g(x);\n`f(x) / g(x)` = f(x) / g(x);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 169 "f := x -> x^2 - 6*x + 13; \+ \ng := x -> x^2 + 9*x - 22; \n`f(x) + g(x)` = f(x) + g(x);\n`f(x) - g( x)` = f(x) - g(x);\n`f(x) * g(x)` = f(x) * g(x);\n`f(x) / g(x)` = f(x) / g(x);" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 51 "2. Arithmetic Opera tions on Functions - Numerically" }}{PARA 0 "" 0 "" {TEXT -1 68 "\nLet s look at how these operations affect the the numerical values.\n" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "f := x -> 2*x ; g := x -> 3* x;\nFuncOps(f, g, 1, 5);\n\n\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "f := x -> 2*x + 3 ; g := x -> -4*x + 20; FuncOps(f, g, 1, 5) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "f := x -> x ; g := x - > 1/x; FuncOps(f, g, 1, 6);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "f := x -> x^2 + 3*x + 2 ; g := x -> x^2 -5*x + 7; FuncOps(f, g , 1, 6);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "f := x -> abs(x -3) ; g := x -> abs(x-5); FuncOps(f, g, 1, 7);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 51 "3. Arithmetic Operations on Functions - Graphically" } }{PARA 0 "" 0 "" {TEXT -1 68 "\nJust for fun, lets look at how these o perations affect the graphs.\n" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 9 " Example 1" }}{PARA 0 "" 0 "" {TEXT -1 111 "\nHere are two functions. W e will graph the functions along with their sums, differences, product , and quotient." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "f := x -> 20; \ng := x -> x;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 394 "r : = y = -30..30:\nA := plot( \{ f(x), g(x) \}, x = -5..5, r, color = bl ack, thickness = 1):\nBp := plot( f(x)+g(x), x = -5..5, r, color = b lue, thickness = 2):\nBm := plot( \{f(x)-g(x), g(x)-f(x)\}, x = -5..5 , r, color = green, thickness = 2):\nCp := plot( f(x)*g(x), x = -5.. 5, r, color = red, thickness = 2):\nCm := plot( f(x)/g(x), x = -5..5, r, color = gold, thickness = 2, discont = true):\n" }}}{PARA 0 "" 0 " " {TEXT -1 104 "\nThe origial functions are in black and thin. The sum is blued, and differences, f-g and g-f are green." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "plots[disp lay](A,Bp,Bm);" }}}{PARA 0 "" 0 "" {TEXT -1 87 "\nThe origial function s are in black and thin. The product is red, and quotient is gold." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "plots[display](A,Cp,Cm);" }} }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{PARA 0 "" 0 "" {TEXT -1 111 "\nHere are two functions. We will graph the functions al ong with their sums, differences, product, and quotient." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "f := x -> 3*x; \ng := x -> 2*x;" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 394 "r := y = -20..20:\nA := pl ot( \{ f(x), g(x) \}, x = -5..5, r, color = black, thickness = 1):\nB p := plot( f(x)+g(x), x = -5..5, r, color = blue, thickness = 2):\nB m := plot( \{f(x)-g(x), g(x)-f(x)\}, x = -5..5, r, color = green, th ickness = 2):\nCp := plot( f(x)*g(x), x = -5..5, r, color = red, thic kness = 2):\nCm := plot( f(x)/g(x), x = -5..5, r, color = gold, thick ness = 2, discont = true):\n" }}}{PARA 0 "" 0 "" {TEXT -1 104 "\nThe o rigial functions are in black and thin. The sum is blued, and differen ces, f-g and g-f are green." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "plots[display](A,Bp,Bm);" }} }{PARA 0 "" 0 "" {TEXT -1 87 "\nThe origial functions are in black and thin. The product is red, and quotient is gold." }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 24 "plots[display](A,Cp,Cm);" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 9 "Example 3" }}{PARA 0 "" 0 "" {TEXT -1 111 "\nHere \+ are two functions. We will graph the functions along with their sums, \+ differences, product, and quotient." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "f := x -> x; \ng := x -> floor(x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 394 "r := y = -10..10:\nA := plot( \{ f(x), g (x) \}, x = -5..5, r, color = black, thickness = 1):\nBp := plot( f(x )+g(x), x = -5..5, r, color = blue, thickness = 2):\nBm := plot( \{f (x)-g(x), g(x)-f(x)\}, x = -5..5, r, color = green, thickness = 2): \nCp := plot( f(x)*g(x), x = -5..5, r, color = red, thickness = 2):\n Cm := plot( f(x)/g(x), x = -5..5, r, color = gold, thickness = 2, dis cont = true):\n" }}}{PARA 0 "" 0 "" {TEXT -1 105 "\nThe origial functi ons are in black and thin. The sum is blued, and differences, f-g and \+ g-f, are green." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 24 "plots[display](A,Bp,Bm);" }}}{PARA 0 "" 0 "" {TEXT -1 87 "\nThe origial functions are in black and thin. The produc t is red, and quotient is gold." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "plots[display](A,Cp,Cm);" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 9 "Example 4" }}{PARA 0 "" 0 "" {TEXT -1 111 "\nHere are two functi ons. We will graph the functions along with their sums, differences, p roduct, and quotient." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "f : = x -> 11*x^2 - 6*x + 13; \ng := x -> 8*x^2 + 19*x - 41;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 394 "r := y = -60..80:\nA := plot( \{ f (x), g(x) \}, x = -5..5, r, color = black, thickness = 1):\nBp := plo t( f(x)+g(x), x = -5..5, r, color = blue, thickness = 2):\nBm := plo t( \{f(x)-g(x), g(x)-f(x)\}, x = -5..5, r, color = green, thickness \+ = 2):\nCp := plot( f(x)*g(x), x = -5..5, r, color = red, thickness = \+ 2):\nCm := plot( f(x)/g(x), x = -5..5, r, color = gold, thickness = 2 , discont = true):\n" }}}{PARA 0 "" 0 "" {TEXT -1 104 "\nThe origial f unctions are in black and thin. The sum is blued, and differences, f-g and g-f are green." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "plots[display](A,Bp,Bm);" }}}{PARA 0 "" 0 "" {TEXT -1 87 "\nThe origial functions are in black and thin. The p roduct is red, and quotient is gold." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "plots[display](A,Cp,Cm);" }}}}}{SECT 0 {PARA 4 "" 0 " " {TEXT -1 53 "4. Composite Functions - Numeric & Algebraic Approach" }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 256 35 " The Idea of Compo site Functions" }{TEXT -1 437 "\n\nTo compose functions means to evalu ate one function, and then take that output as the input for the next \+ function. In a sense, the functions are applied serially. This is not \+ an arithmetic operation. In the operations of addition, subtraction, m ultiplication, and division we looked above, the values of each functi on were computed and then the operation performed. In this case, its a s though one function is embedded into the other.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "f := x -> 2*x + 1; g := x -> 3*x - 7;\n" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "g(1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "`f(g(1))` = f(g(1));" }}}{PARA 0 "" 0 "" {TEXT -1 98 "\nWe find \+ f(g(10)) by computing g(10) = 23, and then computing f(23). Lets try s ome more examples.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "f := x -> 2*x + 1; g := x -> 3*x - 7;\ng( 0 ); `g(0)` = %; `f(g(0))` = f( %%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "f := x -> 13*x + 7; g := x -> 1/(x+2);\ng( 4 ); `g(4)` = %; `f(g(4))` = f(%%);" }}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT 258 44 " The Idea of Composite Functions - Tables" }}{PARA 0 "" 0 "" {TEXT -1 267 "\n\n To compose functions means to evaluate one function, and then take tha t output as the input for the other. In a sense, the functions are app lied serially. This is not an arithmetic operation. In the operations \+ of addition, subtraction, multiplication, and division " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "f:= x-> 3*x; g:= x-> 2*x; FuncComp( f,g, 1,10);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 58 "f:= x-> 4*x - 7; g:= x-> 3*x + 2; FuncComp( f,g, 1,10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "f:= x-> 10* x; g:= x-> x^2 ; FuncComp( f,g, 1,8);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 257 36 " \n Simplifying Composite Functions" }}{PARA 0 "" 0 "" {TEXT -1 93 "\nWhen composing functions, its often necessary to do a little algebr a to simplify the result." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "f := x -> (x+1)^2 ; g := x -> 4 + 7 /x;\nf(g(x)): % = expand(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "f := x -> x^2 + 2 ; g := x -> 5*x + 11;\nf(g(x)): % = expand(%); " }}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 259 27 " Self-Composite \+ Functions" }}{PARA 0 "" 0 "" {TEXT -1 101 "\nWhen you compose two func tions, there is nothing stopping you from composing a function with it self!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "f := x -> 20*x - 33;\nf(f(x));" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 53 "g := x -> (x+1)/(x+2);\ng(g(x)); simplify( exp and(%));" }}}{PARA 0 "" 0 "" {TEXT -1 2 "\n " }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 260 45 "5. Composition is NOT Commuative (in general)" }} {PARA 0 "" 0 "" {TEXT -1 117 "\nAddition and multiplication of functio ns is commutative - which means that a different order gives the same \+ results." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "f := x -> 5*x + 11 ; g := x -> (x-2)/(x+7);\n`(f+g)( 1)`=f(1) + g(1), `(g+f)(1)`=g(1) + f(1);\n`(f*g)(1)`=f(1) * g(1), \+ `(g*f)(1)`=g(1) * f(1);\n" }}}{PARA 0 "" 0 "" {TEXT -1 187 "\nHowever, composition of functions is usually NOT commutative. If you change th e order of composition from f(g(x)) to g(f(x)), you usually get differ ent values. Lets try out some numbers." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "`f(g(1))` = f(g(1)); `g(f(1))` = g(f(1)); " }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "`(f+g)(3)`=f(3) + g(3), \+ `(g+f)(3)`=g(3) + f(3);\n`(f*g)(3)`=f(3) * g(3), `(g*f)(3)`=g(3) * f (3);\n`f(g(3))` = f(g(3)), `g(f(3))` = g(f(3)); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "`(f+g)(0)`=f(0) + g(0), `(g+f)(3 )`=g(0) + f(0);\n`(f*g)(0)`=f(0) * g(0), `(g*f)(3)`=g(0) * f(0);\n`f( g(0))` = f(g(0)), `g(f(0))` = g(f(0)); " }}}{PARA 0 "" 0 "" {TEXT -1 97 "\nCompare these two tables - same two functions, same dom ain. But the final results are different!" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "f:= x-> 10*x; g:= x-> x^2 ; \nFuncComp( f,g, \+ 1,8); FuncComp( g,f, 1,8); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "f:= x-> x^2; g:= x-> x + 10 ; \nFuncComp( f,g, 1,8); \+ FuncComp( g,f, 1,8);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "f: = x-> 10; g:= x-> x^2 ; \nFuncComp( f,g, 1,8); FuncComp( g,f , 1,8);" }}}{PARA 0 "" 0 "" {TEXT -1 38 "\n\nWe can see this algebraic ally also.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 143 "f := x -> 5 *x + 11 ; g := x -> (x-2)/(x+7);\nf(g(x)): `f(g(x))` = %; simplify( e xpand( %%) );\ng(f(x)): `g(f(x))` = %; simplify( expand( %%) );" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "f := x -> x^2 + 10*x - 7 ; \+ g := x -> x^2 -10*x + 7;\nf(g(x)): `f(g(x))` = %; simplify( expand( % %) );\ng(f(x)): `g(f(x))` = %; simplify( expand( %%) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "f := x -> x^3 + x ; g := x -> 1/(x +10);\nf(g(x)): `f(g(x))` = %; simplify( expand( %%) );\ng(f(x)): `g (f(x))` = %; simplify( expand( %%) );" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 56 "6. Composite Functions - Repeated Composition [Optional] " }}{PARA 0 "" 0 "" {TEXT -1 152 "\nAn interesting thing sometimes hap pens when you evaluate a function, then take the output and evaluate i t again with the same function, over and over. " }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 18 "f := x -> x/3 + 7;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 26 "f(0); (f@f)(0); (f@@3)(0);" }}}{PARA 0 "" 0 "" {TEXT -1 106 "\nLook at what happens at 20 iterations of f(x) at x = 0 . The values appear to approach a particular value." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "for k from 1 to 20 do evalf( (f@@k)(0) ); o d;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "f := x -> (x^2) - x + 2/3;\nfor k from 1 to 12 do evalf( (f@@k)(0) ); od;" }}}{PARA 0 "" 0 "" {TEXT -1 87 "\nIn other cases, the values bounce back and forth, or converge to two different values." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "f := x -> (x^2) - 1;\nfor k from 1 to 10 do evalf( (f @@k)(0) ); od;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "f := x -> (x^2) - 9/10;\nfor k from 1 to 12 do evalf( (f@@k)(0) ); od;" }}} {PARA 0 "" 0 "" {TEXT -1 2 "\n " }}}{PARA 0 "" 0 "" {TEXT 264 36 "\n \+ \251 2002 Waterloo Maple Inc " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "0 0" 26 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }