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Chapitre4

Applications

1.Denitionsetexemples

l'ensembled'arriveeouensemblebutdef.

Onnotef:E!Fouf:E!F

x7!f(x).L'ensembleG=f(x;y)2EFjy=f(x)gest appelelegraphedef. 1et2. 32
1 4 3 2 1

Diagrammesagittal

3214
3 2 1

Diagrammecartesien

L'applicationLogarithme:ln:R+!R

x7!ln(x)

L'application:R3!R3

(x;y;z)7!(2x+3y;xy+z;y+5z) p

1:RR!R

(x;y)7!x

L'application\identite":IdE:E!E

x7!x complexes". \L'applicationdeRdansRdenieparf(x)=1=x". \L'applicationfdeniesurZparf(x)=x2"

Compositiondesapplications

f:Df!F etonetudiel'applicationf:R!R x7!1=x. deEdansF.

2.Egalite-Restriction-Prolongement

E=E0;F=F0et8x2E;f(x)=f1(x):

Exemples-Soientf:R!R

x7!cos(x)etf1:R!R x7!2cos2(x=2)1Alors,onaf=f1.

Sionconsideref:R!R

x7!x2,g:R!R+ x7!x2eth:R+!R x7!x2,onobtient troisapplicationsdeuxadeuxdistinctes. f

1=fjE1:

cestroisapplications). l'applicationg:E!F1 etg,sionprendF1=R+: {48{

APPLICATIONS

3.Compositiondesapplications

applicationdeEdansGnoteegfenposant

8x2E;gf(x)=g(f(x)):

Onl'appelleapplicationcomposeedegetf.

sionaf:R!R x7!x2etg:R!R x7!2x,onobtientgf:R!R x7!2x2et fg:R!R

Ona(gf)h=g(fh)(lorsquecelaaunsens).

f

1:E!F1

delangage. gf.A-t-onfg=gf? 2

Calculeretcomparerfgetgf.

4.Familles

n7!unplut^otque u:N!E i7!uietonparlealors naturelle,ondenit: al'undesensemblesAiaumoins: i2IA i=fx2Xj9i2I;x2Aig nentatouslesensemblesAi: i2IA i=fx2Xj8i2I;x2Aig {49{

Familles

X=[ i2IA i

8i;j2I;(i6=j=)Ai\Aj=;)

8i2I;Ai6=;

de[0;+1[: n2NAn.

Quepeut-ondiredelafamille(An)n2N?

2 )CalculerS x2]0;1=2[]x1;x+1[etT x2]0;1=2[]x1;x+1[.

5.Bijection-Injection-Surjection

8y2F;9x2E;y=f(x)

8(x;x0)2E2;(f(x)=f(x0)=)x=x0):

surjective. x festinjective. qu'uneapplicationestinjective. soitenaun.

Exemples-L'applicationl:(R!R

etunseulreelxtelquey=x3. {50{

APPLICATIONS

L'applicationu:(R!R+

injectivecaru(1)=u(1)et16=1:

L'applicationv:(R!R

dessous: 32
1 4 3 2 1 f32 1 3 4 2 1 g 32
1 1 3 2 h 3 2 1 4 3 2 1 k

8y2F;9!x2E;y=f(x)

Remarques-Soitf:E!Funeapplication.

distinctsxetx0deEtelsquef(x)=f(x0). deFquin'aaucunantecedent. y=f(x)n'aaucunesolutiondans[0;1]. {51{

Etudedesbijections

Exercice-1)L'applicationf:R!R+

2

6.Etudedesbijections

Consideronslabijectionl:R!R

x7!x3.L'applicationreciproquedelest l 1:R!R x7!3p x.

Exercice-1)Montrerquel'applicationh:R!R

x7!2x1estbijectiveetdeterminerh1. 2 telleque8x2E;f(x)=g(x).Determinerg1.

1)f1estbijectiveet(f1)1=f,

2)f1f=IdEetff1=IdF

Demonstration:

{52{

APPLICATIONS

pardenitiondef1.D'ouf1f=IdE.

Onfaitdem^emepourmontrerqueff1=IdF.

eneetlapropositionsuivante: f=g1. precedente. i=02 isi l'applicationN:E!N attribuesetN0l'applicationN0:E!M e7!N(e),alorsN0estbijective. {53{

Imagedirecteoureciproque

7.Imagedirecteoureciproque

Onxetoujoursuneapplicationf:E!F.

x2f1(B)()f(x)2B f(A).OnadoncpourtoutelementydeF: y2f(A)()9x2A;y=f(x):

L'ensemblef(E)estaussiappelel'imagedef.

5 3 2 1 4 4 3 2 1

Onaf1(f2g)=;,

f

1(f1g)=f1(f1;2;4g)=f1;2g,

f(f1;4g)=f1;5getl'imagedefest f(f1;2;3;4g)=f1;3;5g: 2 )Soitg:R!R 3 reciproquedeRetcelledef1g. pastresheureuse. 4 f 4 mettretouteslesaccoladesnecessaires. {54{

APPLICATIONS

[x]?

8.Complements

1)festinjective.

2)L'imagedefestl'ensemble[f(a);f(b)].

[a;b]![f(a);f(b)] x7!f(x)etcetteapplicationgestbijective. quelconque. 4

Fabriquezuncontre-exemple.

1)festbijective

2)festinjective

3)festsurjective

4 1) f1;2g!f1;4;6g x7!x2 2) R!R+ x7!x2 3) N!N n7!n+1

EnncetheoremequevousetudierezenMA3:

1)festbijective

2)festinjective

3)festsurjective

{55{

Exercicesd'application

EXERCICESD'APPLICATION

Exercicen1

xettoutydeE,onaith(x+y)=h(x)+h(y).

Exercicen2

f(x)=(1=2xsix2[0;1=2[

0sinong(x)=(0six2[0;1=2[

x1=2sinon

Exercicen3

tellequefh=IdF?

Exercicen4

n2NA netmontrerquela famille(An)n2NformeunepartitiondeE.

Exercicen5

injective,surjective?

Exercicen6

Exercicen7

Soitfl'applicationf:C!C:

z7!1+z2

1)Montrerquefestsurjective.

2)L'applicationfest-elleinjective?

Exercicen8

1-a)Montrerquefn'estpasinjective.

ZZ.

L'applicationhest-ellesurjective?

b)LarestrictiondehaNNest-elleinjective? {56{

APPLICATIONS

Exercicen9

festbijective.

Exercicen10

INDICATIONSETSOLUTIONSSOMMAIRES

Exercicen1

Exercicen2

Lesdeuxapplicationssontdistinctes.

Exercicen3

Exercicen4

E=]0;1[.

Exercicen5

Exercicen6

Exercicen7

siu=1.

1)Doncfestsurjective,

2)etfn'estpasinjective

3)f(R)=fu2Rju>1g.

Exercicen8

1-a)f(0)=f(1)=0et06=1.

entiers. {57{

Exercicesd'application

3)non:h(1;4)=h(3;3)=12et(1;4)6=(3;3).

Exercicen9

Exercicen10

1)Appliquerlesdenitions.

{58{quotesdbs_dbs19.pdfusesText_25

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