20-A \303\201lgebra Das Fun\303\247\303\265es Combinando fun\303\247\303\265es por soma, produto, diferen\303\247a, divis\303\243o e composi\303\247\303\243o.

restart; with(plots):

Warning, the name changecoords has been redefined

FuncOps := proc( f, g, a, b ) local N, M, P, i,j,k, A, p ; A := array( [seq( [ seq( ` `, j = 1..7 ) ], i = a..b+1 ) ]); A[1,1] := `x`; A[1,2] := `f(x)`; A[1,3] := `g(x)`; A[1,4] := `f(x)+g(x)`; A[1,5] := `f(x)-g(x)`; A[1,6] := `f(x)*g(x)`; A[1,7] := `f(x)/g(x)`; for i from a to b do A[i+1, 1] := i; od; for i from a to b do A[i+1, 2] := f(i); od; for i from a to b do A[i+1, 3] := g(i); od; for i from a to b do A[i+1, 4] := f(i)+g(i); od; for i from a to b do A[i+1, 5] := f(i)-g(i); od; for i from a to b do A[i+1, 6] := f(i)*g(i); od; for i from a to b do if( g(i) = 0) then A[i+1, 7] := `undef`; else A[i+1, 7] := f(i)/g(i); fi; od; print(A); end proc: FuncComp := proc( f, g, a, b ) local N, M, P, i,j,k, A, p ; A := array( [seq( [ seq( ` `, j = 1..3 ) ], i = a..b+1 ) ]); A[1,1] := `x`; A[1,2] := `g(x)`; A[1,3] := `f(g(x))`; for i from a to b do A[i+1, 1] := i; od; for i from a to b do A[i+1, 2] := g(i); od; for i from a to b do A[i+1, 3] := f(g(i)); od; print(A); end proc:

As quatro opera\303\247\303\265es aritm\303\251ticas de soma, subtra\303\247\303\243o, multiplica\303\247\303\243o e divis\303\243o podem ser realizadas com fun\303\247\303\265es assim como podem ser realizadas com n\303\272meros e express\303\265es. E \303\251 bem simples. Quando somando ou subtraindo, voc\303\252 pode combinar termos semelhantes. f := x -> 3*x + 2; g := x -> 8*x - 7; `f(x) + g(x)` = f(x) + g(x); `f(x) - g(x)` = f(x) - g(x); `f(x) * g(x)` = f(x) * g(x); `f(x) / g(x)` = f(x) / g(x);

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f := x -> x; g := x -> 1/x; `f(x) + g(x)` = f(x) + g(x); `f(x) - g(x)` = f(x) - g(x); `f(x) * g(x)` = f(x) * g(x); `f(x) / g(x)` = f(x) / g(x);

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f := x -> x^2 - 6*x + 13; g := x -> x^2 + 9*x - 22; `f(x) + g(x)` = f(x) + g(x); `f(x) - g(x)` = f(x) - g(x); `f(x) * g(x)` = f(x) * g(x); `f(x) / g(x)` = f(x) / g(x);

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Vejamos como essas opera\303\247\303\265es afetam os valores num\303\251ricos. f := x -> 2*x ; g := x -> 3*x; FuncOps(f, g, 1, 5);

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f := x -> 2*x + 3 ; g := x -> -4*x + 20; FuncOps(f, g, 1, 5);

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f := x -> x ; g := x -> 1/x; FuncOps(f, g, 1, 6);

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f := x -> x^2 + 3*x + 2 ; g := x -> x^2 -5*x + 7; FuncOps(f, g, 1, 6);

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f := x -> abs(x-3) ; g := x -> abs(x-5); FuncOps(f, g, 1, 7);

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Apenas por curiosidade, vamos ver como essas opera\303\247\303\265es afetam os gr\303\241ficos.

Temos aqui duas fun\303\247\303\265es. Vamos ver os gr\303\241ficos dessas fun\303\247\303\265es ao longo de suas somas, diferen\303\247as, produtos e quocientes. f := x -> 20; g := x -> x;

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r := y = -30..30: A := plot( { f(x), g(x) }, x = -5..5, r, color = black, thickness = 1): Bp := plot( f(x)+g(x), x = -5..5, r, color = blue, thickness = 2): Bm := plot( {f(x)-g(x), g(x)-f(x)}, x = -5..5, r, color = green, thickness = 2): Cp := plot( f(x)*g(x), x = -5..5, r, color = red, thickness = 2): Cm := plot( f(x)/g(x), x = -5..5, r, color = gold, thickness = 2, discont = true): As fun\303\247\303\265es originais est\303\243o em preto fino. As somas em azul, e as diferen\303\247as, f-g e g-f est\303\243o verdes. plots[display](A,Bp,Bm);

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As fun\303\247\303\265es originais est\303\243o em preto fino. O produto \303\251 vermelho, e o quociente dourado. plots[display](A,Cp,Cm);

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Temos aqui duas fun\303\247\303\265es. Vamos ver os gr\303\241ficos dessas fun\303\247\303\265es ao longo de suas somas, diferen\303\247\303\243s, produtos e quocientes. f := x -> 3*x; g := x -> 2*x;

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r := y = -20..20: A := plot( { f(x), g(x) }, x = -5..5, r, color = black, thickness = 1): Bp := plot( f(x)+g(x), x = -5..5, r, color = blue, thickness = 2): Bm := plot( {f(x)-g(x), g(x)-f(x)}, x = -5..5, r, color = green, thickness = 2): Cp := plot( f(x)*g(x), x = -5..5, r, color = red, thickness = 2): Cm := plot( f(x)/g(x), x = -5..5, r, color = gold, thickness = 2, discont = true): As fun\303\247\303\265es originais est\303\243o em preto fino. As somas em azul, e as diferen\303\247as, f-g e g-f est\303\243o verdes. plots[display](A,Bp,Bm);

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As fun\303\247\303\265es originais est\303\243o em preto fino. O produto \303\251 vermelho, e o quociente dourado. plots[display](A,Cp,Cm);

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Temos aqui duas fun\303\247\303\265es. Vamos ver os gr\303\241ficos dessas fun\303\247\303\265es ao longo de suas somas, diferen\303\247as, produtos e quocientes. f := x -> x; g := x -> floor(x);

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r := y = -10..10: A := plot( { f(x), g(x) }, x = -5..5, r, color = black, thickness = 1): Bp := plot( f(x)+g(x), x = -5..5, r, color = blue, thickness = 2): Bm := plot( {f(x)-g(x), g(x)-f(x)}, x = -5..5, r, color = green, thickness = 2): Cp := plot( f(x)*g(x), x = -5..5, r, color = red, thickness = 2): Cm := plot( f(x)/g(x), x = -5..5, r, color = gold, thickness = 2, discont = true): As fun\303\247\303\265es originais est\303\243o em preto fino. As somas em azul, e as diferen\303\247as, f-g e g-f est\303\243o verdes. plots[display](A,Bp,Bm);

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As fun\303\247\303\265es originais est\303\243o em preto fino. O produto \303\251 vermelho, e o quociente dourado. plots[display](A,Cp,Cm);

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Temos aqui duas fun\303\247\303\265es. Vamos ver os gr\303\241ficos dessas fun\303\247\303\265es ao longo de suas somas, diferen\303\247as, produtos e quocientes. f := x -> 11*x^2 - 6*x + 13; g := x -> 8*x^2 + 19*x - 41;

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r := y = -60..80: A := plot( { f(x), g(x) }, x = -5..5, r, color = black, thickness = 1): Bp := plot( f(x)+g(x), x = -5..5, r, color = blue, thickness = 2): Bm := plot( {f(x)-g(x), g(x)-f(x)}, x = -5..5, r, color = green, thickness = 2): Cp := plot( f(x)*g(x), x = -5..5, r, color = red, thickness = 2): Cm := plot( f(x)/g(x), x = -5..5, r, color = gold, thickness = 2, discont = true): As fun\303\247\303\265es originais est\303\243o em preto fino. As somas em azul, e as diferen\303\247as, f-g e g-f est\303\243o verdes. plots[display](A,Bp,Bm);

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As fun\303\247\303\265es originais est\303\243o em preto fino. O produto \303\251 vermelho, e o quociente dourado. plots[display](A,Cp,Cm);

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A Id\303\251ia de Fun\303\247\303\265es Compostas Compor uma fun\303\247\303\243o significa calcular uma fun\303\247\303\243o e ent\303\243o usar esse resultado como a entrada da pr\303\263xima fun\303\247\303\243o. De certo modo, as fun\303\247\303\265es s\303\243o aplicadas em s\303\251rie. Isso n\303\243o \303\251 uma opera\303\247\303\243o aritm\303\251tica. Nas opera\303\247\303\265es de adi\303\247\303\243o, subtra\303\247\303\243o, multiplica\303\247\303\243o e divis\303\243o que estudamos acima, os valores de cada fun\303\247\303\243o eram computados e a opera\303\247\303\243o realizada. Neste caso, \303\251 como se uma fun\303\247\303\243o estivesse em fun\303\247\303\243o da outra. f := x -> 2*x + 1; g := x -> 3*x - 7;

NiM+JSJmR2YqNiMlInhHNiI2JCUpb3BlcmF0b3JHJSZhcnJvd0c2IiwmKiYiIiMiIiIlInhHIiIiIiIiIiIiIiIiNiI2IjYiNiM+JSJnR2YqNiMlInhHNiI2JCUpb3BlcmF0b3JHJSZhcnJvd0c2IiwmKiYiIiQiIiIlInhHIiIiIiIiIiIoISIiNiI2IjYi

g(1);

NiMhIiU=

f(%);

NiMhIig=

`f(g(1))` = f(g(1));

NiMvJShmKGcoMSkpRyEiKA==

Encontramos f(g(10)) computando g(10) = 23 e ent\303\243o calculando f(23). Vamos tentar mais alguns exemplos. f := x -> 2*x + 1; g := x -> 3*x - 7; g( 0 ); `g(0)` = %; `f(g(0))` = f(%%);

NiM+JSJmR2YqNiMlInhHNiI2JCUpb3BlcmF0b3JHJSZhcnJvd0c2IiwmKiYiIiMiIiIlInhHIiIiIiIiIiIiIiIiNiI2IjYiNiM+JSJnR2YqNiMlInhHNiI2JCUpb3BlcmF0b3JHJSZhcnJvd0c2IiwmKiYiIiQiIiIlInhHIiIiIiIiIiIoISIiNiI2IjYiNiMhIig=NiMvJSVnKDApRyEiKA==NiMvJShmKGcoMCkpRyEjOA==

f := x -> 13*x + 7; g := x -> 1/(x+2); g( 4 ); `g(4)` = %; `f(g(4))` = f(%%);

NiM+JSJmR2YqNiMlInhHNiI2JCUpb3BlcmF0b3JHJSZhcnJvd0c2IiwmKiYiIzgiIiIlInhHIiIiIiIiIiIoIiIiNiI2IjYiNiM+JSJnR2YqNiMlInhHNiI2JCUpb3BlcmF0b3JHJSZhcnJvd0c2IiomIiIiIiIiLCYlInhHIiIiIiIjIiIiISIiNiI2IjYiNiMjIiIiIiInNiMvJSVnKDQpRyMiIiIiIic=NiMvJShmKGcoNCkpRyMiI2IiIic=

A Id\303\251ia de Fun\303\247\303\265es Compostas - TabelasCompor fun\303\247o\303\265es significa avaliar uma fun\303\247\303\243o e apanhar o resultado e colocar como valor inicial para uma outra fun\303\247\303\243o. Isto \303\251, as fun\303\247\303\265es s\303\243o empregadas seriamente. Isto n\303\243o \303\251 uma opera\303\247\303\243o aritm\303\251tica. Numa opera\303\247\303\243o de adi\303\247\303\243o, subtra\303\247\303\243o, multiplica\303\247\303\243o e divis\303\243o. f:= x->3*x; g:= x-> 2*x; FuncComp( f,g, 1,10);

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f:= x-> 4*x - 7; g:= x-> 3*x + 2; FuncComp( f,g, 1,10);

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f:= x-> 10*x; g:= x-> x^2 ; FuncComp( f,g, 1,8);

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Simplificando Fun\303\247\303\243o Composta Ao compor fun\303\247\303\265es \303\251 comum que se precise de um pouco de algebra para simplificar o resultado. f := x -> (x+1)^2 ; g := x -> 4 + 7/x; f(g(x)): % = expand(%);

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f := x -> x^2 + 2 ; g := x -> 5*x + 11; f(g(x)): % = expand(%);

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Fun\303\247\303\265es Auto Compostas Quando se comp\303\265e duas fun\303\247\303\265es, n\303\243o h\303\241 nada que impe\303\247a que se componha a fun\303\247\303\243o consigo mesma! f := x -> 20*x - 33; f(f(x));

NiM+JSJmR2YqNiMlInhHNiI2JCUpb3BlcmF0b3JHJSZhcnJvd0c2IiwmKiYiIz8iIiIlInhHIiIiIiIiIiNMISIiNiI2IjYiNiMsJiomIiQrJSIiIiUieEciIiIiIiIiJCRwISIi

g := x -> (x+1)/(x+2); g(g(x)); simplify( expand(%));

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Adi\303\247\303\243o e multiplica\303\247\303\243o de fun\303\247\303\265es s\303\243o comutativas - o que significa que a ordem dos fatores n\303\243o altera o resultado. f := x -> 5*x + 11 ; g := x -> (x-2)/(x+7); `(f+g)(1)`=f(1) + g(1), `(g+f)(1)`=g(1) + f(1); `(f*g)(1)`=f(1) * g(1), `(g*f)(1)`=g(1) * f(1);

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Entretanto, a composi\303\247\303\243o de fun\303\247\303\265es geralmente N\303\203O \303\251 comutativa. Se voc\303\252 mudar a ordem de composi\303\247\303\243o de f(g(x)) para g(f(x)), voc\303\252 geralmente obt\303\251m resultados diferentes. Vamos tentar alguns n\303\272meros. `f(g(1))` = f(g(1)); `g(f(1))` = g(f(1));

NiMvJShmKGcoMSkpRyMiIyQpIiIpNiMvJShnKGYoMSkpRyMiIzkiI0I=

`(f+g)(3)`=f(3) + g(3), `(g+f)(3)`=g(3) + f(3); `(f*g)(3)`=f(3) * g(3), `(g*f)(3)`=g(3) * f(3); `f(g(3))` = f(g(3)), `g(f(3))` = g(f(3));

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`(f+g)(0)`=f(0) + g(0), `(g+f)(3)`=g(0) + f(0); `(f*g)(0)`=f(0) * g(0), `(g*f)(3)`=g(0) * f(0); `f(g(0))` = f(g(0)), `g(f(0))` = g(f(0));

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Compare essas duas tabelas - as mesmas duas fun\303\247\303\265es, mesmo dom\303\255nio. Mas os resultados finais s\303\243o diferentes! f:= x-> 10*x; g:= x-> x^2 ; FuncComp( f,g, 1,8); FuncComp( g,f, 1,8);

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f:= x-> x^2; g:= x-> x + 10 ; FuncComp( f,g, 1,8); FuncComp( g,f, 1,8);

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f:= x-> 10; g:= x-> x^2 ; FuncComp( f,g, 1,8); FuncComp( g,f, 1,8);

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Tamb\303\251m podemos ver isso algebricamente. f := x -> 5*x + 11 ; g := x -> (x-2)/(x+7); f(g(x)): `f(g(x))` = %; simplify( expand( %%) ); g(f(x)): `g(f(x))` = %; simplify( expand( %%) );

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f := x -> x^2 + 10*x - 7 ; g := x -> x^2 -10*x + 7; f(g(x)): `f(g(x))` = %; simplify( expand( %%) ); g(f(x)): `g(f(x))` = %; simplify( expand( %%) );

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f := x -> x^3 + x ; g := x -> 1/(x+10); f(g(x)): `f(g(x))` = %; simplify( expand( %%) ); g(f(x)): `g(f(x))` = %; simplify( expand( %%) );

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Certas vezes, acontece algo interessante quando se calcula uma fun\303\247\303\243o toma-se esse resultado e calcula-se novamente com a mesma fun\303\247\303\243o, e de novo e de novo. f := x -> x/3 + 7;

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f(0); (f@f)(0); (f@@3)(0);

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Veja o que acontece com 20 repeti\303\247\303\265es de f(x) com x = 0. Os valores parecem se aproximar de um valor em particular. for k from 1 to 20 do evalf( (f@@k)(0) ); od;

NiMkIiIoIiIhNiMkIitMTExMJCohIio=NiMkIis2NjY2NSEiKQ==NiMkIitQcS5QNSEiKQ==NiMkIis3IXpjLyIhIik=NiMkIityJ2YmWzUhIik=NiMkIishKik+Jlw1ISIpNiMkIitqKlIpXDUhIik=NiMkIithbSUqXDUhIik=NiMkIis9QSkqXDUhIik=NiMkIit0UyoqXDUhIik=NiMkIitDISkqKlw1ISIpNiMkIitUJCoqKlw1ISIpNiMkIisheSoqKlw1ISIpNiMkIitGKioqKlw1ISIpNiMkIit3KioqKlw1ISIpNiMkIisjKioqKipcNSEiKQ==NiMkIisoKioqKipcNSEiKQ==NiMkIisqKioqKipcNSEiKQ==NiMkIisrKytdNSEiKQ==

f := x -> (x^2) - x + 2/3; for k from 1 to 12 do evalf( (f@@k)(0) ); od;

NiM+JSJmR2YqNiMlInhHNiI2JCUpb3BlcmF0b3JHJSZhcnJvd0c2IiwoKiQpJSJ4RyIiIyIiIiIiIiUieEchIiIjIiIjIiIkIiIiNiI2IjYiNiMkIitubW1tbSEjNQ==NiMkIitXV1dXVyEjNQ==NiMkIitrM2AoPiUhIzU=NiMkIitRQjFKVSEjNQ==NiMkIit6SnpEVSEjNQ==NiMkIisoRzFtQSUhIzU=NiMkIitdL1tFVSEjNQ==NiMkIis7KipcRVUhIzU=NiMkIiswcFxFVSEjNQ==NiMkIitydFxFVSEjNQ==NiMkIispSChcRVUhIzU=NiMkIis1dFxFVSEjNQ==

em outros casos, os valores v\303\243o e voltam ou convergem para dois valores diferentes. f := x -> (x^2) - 1; for k from 1 to 10 do evalf( (f@@k)(0) ); od;

NiM+JSJmR2YqNiMlInhHNiI2JCUpb3BlcmF0b3JHJSZhcnJvd0c2IiwmKiQpJSJ4RyIiIyIiIiIiIiIiIiEiIjYiNiI2Ig==NiMkISIiIiIhNiMkIiIhIiIhNiMkISIiIiIhNiMkIiIhIiIhNiMkISIiIiIhNiMkIiIhIiIhNiMkISIiIiIhNiMkIiIhIiIhNiMkISIiIiIhNiMkIiIhIiIh

f := x -> (x^2) - 9/10; for k from 1 to 12 do evalf( (f@@k)(0) ); od;

NiM+JSJmR2YqNiMlInhHNiI2JCUpb3BlcmF0b3JHJSZhcnJvd0c2IiwmKiQpJSJ4RyIiIyIiIiIiIiMiIioiIzUhIiI2IjYiNiI=NiMkISsrKysrISohIzU=NiMkISsrKysrISohIzY=NiMkISsrKys+KikhIzU=NiMkISsrUjlYNSEjNQ==NiMkIStCdXchKikpISM1NiMkIStqYVUmNCIhIzU=NiMkISswVishKSkpISM1NiMkISthQmI5NiEjNQ==NiMkISswdHh2KSkhIzU=NiMkIStCeDBBNiEjNQ==NiMkIStaJyk0dSkpISM1NiMkIStAdC5ENiEjNQ==