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În cele de mai jos sunt prezentate câteva funcÅ£ii deosebite, folosite în
matematica de liceu, însoÅ£ite de definiÅ£iile ÅŸi proprietăţile lor esenÅ£iale.
2) APLICATIA-1
Data publicării : 24.08.2010Suport teoretic:
Functia parte intreaga, graficul unei functii, intersectia a doua multimi, sistem de inecuatii de gradul intai.
Enunt:
Sa se gaseasca intersectia graficelor functiilor reale de variabila reala, definite prin legile:
![f,g:{\mathbb{R}}\rightarrow{\mathbb{R}},\;f(x)=[\frac{2-x}{4}],\;g(x)=\frac{1}{2}x-\frac{1}{2}.](http://s.wordpress.com/latex.php?latex=f%2Cg%3A%7B%5Cmathbb%7BR%7D%7D%5Crightarrow%7B%5Cmathbb%7BR%7D%7D%2C%5C%3Bf%28x%29%3D%5B%5Cfrac%7B2-x%7D%7B4%7D%5D%2C%5C%3Bg%28x%29%3D%5Cfrac%7B1%7D%7B2%7Dx-%5Cfrac%7B1%7D%7B2%7D.&bg=FFFFFF&fg=000000&s=0 "f,g:{\mathbb{R}}\rightarrow{\mathbb{R}},\;f(x)=[\frac{2-x}{4}],\;g(x)=\frac{1}{2}x-\frac{1}{2}.")
Raspuns:
1) TEORIE
Data publicării : 13.03.2009
Functia modul (valoare absoluta)
este definită prin:
![f(x)=|x|=\begin{cases}-x,x\in{(-\infty,0]}\\x,x\in{(0,+\infty)}\end{cases}\;sau](http://s.wordpress.com/latex.php?latex=f%28x%29%3D%7Cx%7C%3D%5Cbegin%7Bcases%7D-x%2Cx%5Cin%7B%28-%5Cinfty%2C0%5D%7D%5C%5Cx%2Cx%5Cin%7B%280%2C%2B%5Cinfty%29%7D%5Cend%7Bcases%7D%5C%3Bsau&bg=FFFFFF&fg=000000&s=0 "f(x)=|x|=\begin{cases}-x,x\in{(-\infty,0]}\\x,x\in{(0,+\infty)}\end{cases}\;sau")
Functia parte intreaga
este definită prin:
![f:{\mathbb{R}}\rightarrow{\mathbb{Z}},f(x)=[x],](http://s.wordpress.com/latex.php?latex=f%3A%7B%5Cmathbb%7BR%7D%7D%5Crightarrow%7B%5Cmathbb%7BZ%7D%7D%2Cf%28x%29%3D%5Bx%5D%2C&bg=FFFFFF&fg=000000&s=0 "f:{\mathbb{R}}\rightarrow{\mathbb{Z}},f(x)=[x],")
unde prin [x] înÅ£elegem cel mai mare număr întreg n, care este mai mic sau egal cu x, adică [x]=n, dacă
Observatii:
![[x]\leq{x}<{[x]+1},\forall{x}\in{\mathbb{R}}.](http://s.wordpress.com/latex.php?latex=%5Bx%5D%5Cleq%7Bx%7D%3C%7B%5Bx%5D%2B1%7D%2C%5Cforall%7Bx%7D%5Cin%7B%5Cmathbb%7BR%7D%7D.&bg=FFFFFF&fg=000000&s=0 "[x]\leq{x}<{[x]+1},\forall{x}\in{\mathbb{R}}.")
![{x-1}<{[x]}\leq{x},\forall{x}\in{\mathbb{R}}.](http://s.wordpress.com/latex.php?latex=%7Bx-1%7D%3C%7B%5Bx%5D%7D%5Cleq%7Bx%7D%2C%5Cforall%7Bx%7D%5Cin%7B%5Cmathbb%7BR%7D%7D.&bg=FFFFFF&fg=000000&s=0 "{x-1}<{[x]}\leq{x},\forall{x}\in{\mathbb{R}}.")
![f(x)={[x]}=\begin{cases}\cdots\\-k,x\in{[-k,-k+1)}\\\cdots\\-3,x\in{[-3,-2)}\\-2,x\in{[-2,-1)}\\-1,x\in{[-1,0)}\\{0},x\in{[0,1)}\\1,x\in{[1,2)}\\2,x\in{[2,3)}\\3,x\in{[3,4)}\\\cdots\\k,x\in{[k,k+1)}\\\cdots\end{cases},k\in{\mathbb{N}}.](http://s.wordpress.com/latex.php?latex=f%28x%29%3D%7B%5Bx%5D%7D%3D%5Cbegin%7Bcases%7D%5Ccdots%5C%5C-k%2Cx%5Cin%7B%5B-k%2C-k%2B1%29%7D%5C%5C%5Ccdots%5C%5C-3%2Cx%5Cin%7B%5B-3%2C-2%29%7D%5C%5C-2%2Cx%5Cin%7B%5B-2%2C-1%29%7D%5C%5C-1%2Cx%5Cin%7B%5B-1%2C0%29%7D%5C%5C%7B0%7D%2Cx%5Cin%7B%5B0%2C1%29%7D%5C%5C1%2Cx%5Cin%7B%5B1%2C2%29%7D%5C%5C2%2Cx%5Cin%7B%5B2%2C3%29%7D%5C%5C3%2Cx%5Cin%7B%5B3%2C4%29%7D%5C%5C%5Ccdots%5C%5Ck%2Cx%5Cin%7B%5Bk%2Ck%2B1%29%7D%5C%5C%5Ccdots%5Cend%7Bcases%7D%2Ck%5Cin%7B%5Cmathbb%7BN%7D%7D.&bg=FFFFFF&fg=000000&s=0 "f(x)={[x]}=\begin{cases}\cdots\\-k,x\in{[-k,-k+1)}\\\cdots\\-3,x\in{[-3,-2)}\\-2,x\in{[-2,-1)}\\-1,x\in{[-1,0)}\\{0},x\in{[0,1)}\\1,x\in{[1,2)}\\2,x\in{[2,3)}\\3,x\in{[3,4)}\\\cdots\\k,x\in{[k,k+1)}\\\cdots\end{cases},k\in{\mathbb{N}}.")
Functia parte fractionara (zecimala):
![\begin{Bmatrix}x\end{Bmatrix}={x-[x]}.](http://s.wordpress.com/latex.php?latex=%5Cbegin%7BBmatrix%7Dx%5Cend%7BBmatrix%7D%3D%7Bx-%5Bx%5D%7D.&bg=FFFFFF&fg=000000&s=0 "\begin{Bmatrix}x\end{Bmatrix}={x-[x]}.")
CATEGORII :
-
1. BREVIAR TEORETIC
- 1.1. ELEMENTE DE LOGICA MATEMATICA (2)
- 1.2. MULTIMI NUMERICE (2)
- 1.3. NUMERE REALE (3)
- 1.4. IDENTITATI REMARCABILE (2)
- 1.5. INEGALITATI (2)
- 1.6. NUMERE COMPLEXE (4)
- 1.7. PROGRESII (2)
- 1.8. COMBINATORICA (3)
- 1.9. LOGARITMI (2)
- 1.10. POLINOAME CU COEFICIENTI REALI (2)
- 1.11. POLINOAME CU COEFICIENTI COMPLECSI (2)
- 1.12. ECUATII ALGEBRICE (2)
- 1.13. PROBABILITATI (2)
- 1.14. PERMUTARI (2)
- 1.15. MATRICE (2)
- 1.16. DETERMINANTI (2)
- 1.17. CLASE DE RESTURI modulo n (2)
- 1.18. GRUPURI (3)
- 1.19. SISTEME DE ECUATII LINIARE (2)
- 1.20. INELE SI CORPURI (2)
- 1.21. FUNCTII - generalitati (2)
- 1.22. FUNCTII ELEMENTARE (2)
- 1.23. FUNCTII SPECIALE (2)
- 1.24. LIMITE DE SIRURI (2)
- 1.25. LIMITE DE FUNCTII (2)
- 1.26. FUNCTII CONTINUE (2)
- 1.27. FUNCTII DERIVABILE (2)
- 1.28. PRIMITIVE (2)
- 1.29. PROPRIETATI ALE FUNCTIILOR DERIVABILE (2)
- 1.30. INTEGRALE DEFINITE (2)
- 1.31. APLICATII ALE INTEGRALEI DEFINITE (2)
- 1.32. VECTORI (2)
- 1.33. GEOMETRIE SINTETICA IN PLAN (3)
- 1.34. TRIGONOMETRIE (2)
- 1.35. GEOMETRIE SINTETICA IN SPATIU (4)
- 1.36. GEOMETRIE ANALITICA IN PLAN (6)
- 1.37. GEOMETRIE ANALITICA IN SPATIU (2)
- 2. CUM ABORDAM O PROBLEMA? (1)
- 3. ALGORITMI IN MATEMATICA DE LICEU
- 4. PROBLEME DIVERSE CU REZOLVARI COMPLETE. (23)
- 5. ALGEBRA - aplicatii
- 6. PROBABILITATI-aplicatii (10)
- 7. GEOMETRIE - aplicatii
- 8. TRIGONOMETRIE - aplicatii
- 9. ANALIZA - aplicatii
- 10. UNDE ESTE GRESEALA ?
- 11. PROBLEME DISTRACTIVE (8)