Informaţii, definiţii, teoreme, formule, exerciţii şi probleme rezolvate din matematica de liceu. RSS/XML
În cele ce urmează va fi prezentată în mod succint teoria privind
inversabilitatea funcţiilor, precum şi toate funcţiile elementare
(sau restricţiile lor), care sunt bijective, alături de inversele acestora, iar în
final câteva tipuri de exerciţii aplicative.
TEORIE- definiţii
Data publicarii: 13.12.2010Definitie:
O functie f:A - > B este inversabila, daca exista o functie g:B - > A, astfel incat
f o g = 1B si g o f = 1A, unde 1M:M - >M, 1M(x) = x, oricare ar fi x din M, se numeste
aplicatia identica a multimii M.
In cazul particular A = B, are loc egalitatea: f o g = g o f = 1A.
Observatii:
- In mod obisnuit, inversa functiei f, cand exista, se noteaza
f^{-1};
- În cazul cand exista, se arata ca functia g (inversa functiei f) este unica;
- In aceste conditii, are loc echivalenta
y = f(x) <=> x = g(y),
unde x parcurge domeniul de definitie A al functiei f, iar y, imaginea lui x prin functia f,
parcurge domeniul de definitie B al functiei g (codomeniul functiei f);
- Graficele unei functii si al inversei acesteia (cand exista!) sunt simetrice fata de prima bisectoare.
FUNCTII ELEMENTARE INVERSABILE
Data publicarii: 18.12.2010- Functia de gradul intai, inversabila pe R:
Definitie:
f:R - > R, f(x) = y = ax + b, unde a, b sunt numere reale, a nenul.
Inversa (tot functie de gradul intai):
- Functia de gradul al doilea, inversabila, separat, pe intervalele:
I1 = (- oo,- b/2a] si I2 = [- b/2a,+ oo):
Definitii:
1) f:(- 00,- b/2a] - > (- 00,- Δ/4a], f(x) = y = ax² + bx + c, a < 0, b, c € R.
Inversa (functie irationala):
![f^{-1}:{(-\infty,-\frac{\Delta}{4a}]}\rightarrow{(-\infty,-\frac{b}{2a}]},\;{f^{-1}}(y)=x=\frac{-b-\sqrt{b^2-4ac+4ay}}{2a}](http://s.wordpress.com/latex.php?latex=f%5E%7B-1%7D%3A%7B%28-%5Cinfty%2C-%5Cfrac%7B%5CDelta%7D%7B4a%7D%5D%7D%5Crightarrow%7B%28-%5Cinfty%2C-%5Cfrac%7Bb%7D%7B2a%7D%5D%7D%2C%5C%3B%7Bf%5E%7B-1%7D%7D%28y%29%3Dx%3D%5Cfrac%7B-b-%5Csqrt%7Bb%5E2-4ac%2B4ay%7D%7D%7B2a%7D&bg=FFFFFF&fg=000000&s=0 "f^{-1}:{(-\infty,-\frac{\Delta}{4a}]}\rightarrow{(-\infty,-\frac{b}{2a}]},\;{f^{-1}}(y)=x=\frac{-b-\sqrt{b^2-4ac+4ay}}{2a}")
2) f:(- 00,- b/2a] - > (- Δ/4a, + 00), f(x) = y = ax² + bx + c, a > 0, b, c € R.
Inversa:
![f^{-1}:{[-\frac{\Delta}{4a},+\infty)}\rightarrow{(-\infty,-\frac{b}{2a},]},\;{f^{-1}}(y)=x=\frac{-b-\sqrt{b^2-4ac+4ay}}{2a}](http://s.wordpress.com/latex.php?latex=f%5E%7B-1%7D%3A%7B%5B-%5Cfrac%7B%5CDelta%7D%7B4a%7D%2C%2B%5Cinfty%29%7D%5Crightarrow%7B%28-%5Cinfty%2C-%5Cfrac%7Bb%7D%7B2a%7D%2C%5D%7D%2C%5C%3B%7Bf%5E%7B-1%7D%7D%28y%29%3Dx%3D%5Cfrac%7B-b-%5Csqrt%7Bb%5E2-4ac%2B4ay%7D%7D%7B2a%7D&bg=FFFFFF&fg=000000&s=0 "f^{-1}:{[-\frac{\Delta}{4a},+\infty)}\rightarrow{(-\infty,-\frac{b}{2a},]},\;{f^{-1}}(y)=x=\frac{-b-\sqrt{b^2-4ac+4ay}}{2a}")
3) f:[- b/2a, + 00) - > (- 00, - Δ/4a], f(x) = y = ax² + bx + c, a < 0, b, c € R.
Inversa:
![f^{-1}:{(-\infty,-\frac{\Delta}{4a}]}\rightarrow{[-\frac{b}{2a},+\infty)},\;{f^{-1}}(y)=x=\frac{-b+\sqrt{b^2-4ac+4ay}}{2a}.](http://s.wordpress.com/latex.php?latex=f%5E%7B-1%7D%3A%7B%28-%5Cinfty%2C-%5Cfrac%7B%5CDelta%7D%7B4a%7D%5D%7D%5Crightarrow%7B%5B-%5Cfrac%7Bb%7D%7B2a%7D%2C%2B%5Cinfty%29%7D%2C%5C%3B%7Bf%5E%7B-1%7D%7D%28y%29%3Dx%3D%5Cfrac%7B-b%2B%5Csqrt%7Bb%5E2-4ac%2B4ay%7D%7D%7B2a%7D.&bg=FFFFFF&fg=000000&s=0 "f^{-1}:{(-\infty,-\frac{\Delta}{4a}]}\rightarrow{[-\frac{b}{2a},+\infty)},\;{f^{-1}}(y)=x=\frac{-b+\sqrt{b^2-4ac+4ay}}{2a}.")
4) f:[- b/2a,+ 00) - > [- Δ/4a,+ 00), f(x) = y = ax² + bx + c, a > 0, b, c € R.
Inversa:
EXERCITIUL 1
Data publicarii: 14.12.2010Suport teoretic:
Functia radical, rezolvarea unei ecuatii irationale, aflarea functiei inverse.
Enunt:
Se da functia f:R - > R, f(x) = y = ![\sqrt[3]{8x^3+1}.](http://s.wordpress.com/latex.php?latex=%5Csqrt%5B3%5D%7B8x%5E3%2B1%7D.&bg=FFFFFF&fg=000000&s=0 "\sqrt[3]{8x^3+1}.")
Sa se demonstreze ca f este inversabila si apoi sa se determine inversa sa.
Raspuns:
![f^{-1}:{\mathbb{R}}\rightarrow{\mathbb{R}},\;{f^{-1}}(y)=x={\frac{1}{2}}\cdot{\sqrt[3]{y^3-1}}.](http://s.wordpress.com/latex.php?latex=f%5E%7B-1%7D%3A%7B%5Cmathbb%7BR%7D%7D%5Crightarrow%7B%5Cmathbb%7BR%7D%7D%2C%5C%3B%7Bf%5E%7B-1%7D%7D%28y%29%3Dx%3D%7B%5Cfrac%7B1%7D%7B2%7D%7D%5Ccdot%7B%5Csqrt%5B3%5D%7By%5E3-1%7D%7D.&bg=FFFFFF&fg=000000&s=0 "f^{-1}:{\mathbb{R}}\rightarrow{\mathbb{R}},\;{f^{-1}}(y)=x={\frac{1}{2}}\cdot{\sqrt[3]{y^3-1}}.")
EXERCITIUL 2
Data publicarii: 22.12.2010Suport teoretic:
Inecuatii transcendente, functii inverse, arcsinus.
Enunt:
Sa se rezolve in multimea numerelor reale inecuatia:
Raspuns:
![x\in{[-sin{\frac{\sqrt{3}}{2}},sin{\frac{1}{2}}]}.](http://s.wordpress.com/latex.php?latex=x%5Cin%7B%5B-sin%7B%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7D%7D%2Csin%7B%5Cfrac%7B1%7D%7B2%7D%7D%5D%7D.&bg=FFFFFF&fg=000000&s=0 "x\in{[-sin{\frac{\sqrt{3}}{2}},sin{\frac{1}{2}}]}.")
EXERCITIUL 3
Data publicarii: 23.12.2010Suport teoretic:
Functie bijectiva, inversa unei functii bijective, identitati trigonometrice remarcabile.
Enunt:
Sa se demonstreze ca functia
![f:{[\frac{\pi}{4},\frac{5\pi}{4}]}\rightarrow{[-\sqrt{2},\sqrt{2}]},](http://s.wordpress.com/latex.php?latex=f%3A%7B%5B%5Cfrac%7B%5Cpi%7D%7B4%7D%2C%5Cfrac%7B5%5Cpi%7D%7B4%7D%5D%7D%5Crightarrow%7B%5B-%5Csqrt%7B2%7D%2C%5Csqrt%7B2%7D%5D%7D%2C&bg=FFFFFF&fg=000000&s=0 "f:{[\frac{\pi}{4},\frac{5\pi}{4}]}\rightarrow{[-\sqrt{2},\sqrt{2}]},")
f(x) = sinx + cosx,
este bijectiva si sa se determine inversa sa.
Raspuns:
![{f^{-1}}:{[-\sqrt{2},\sqrt{2}]}\rightarrow{[\frac{\pi}{4},\frac{5\pi}{4}]},\;{f^{-1}}(x)=\frac{\pi}{4}+{arccos}{\frac{x}{\sqrt{2}}}.](http://s.wordpress.com/latex.php?latex=%7Bf%5E%7B-1%7D%7D%3A%7B%5B-%5Csqrt%7B2%7D%2C%5Csqrt%7B2%7D%5D%7D%5Crightarrow%7B%5B%5Cfrac%7B%5Cpi%7D%7B4%7D%2C%5Cfrac%7B5%5Cpi%7D%7B4%7D%5D%7D%2C%5C%3B%7Bf%5E%7B-1%7D%7D%28x%29%3D%5Cfrac%7B%5Cpi%7D%7B4%7D%2B%7Barccos%7D%7B%5Cfrac%7Bx%7D%7B%5Csqrt%7B2%7D%7D%7D.&bg=FFFFFF&fg=000000&s=0 "{f^{-1}}:{[-\sqrt{2},\sqrt{2}]}\rightarrow{[\frac{\pi}{4},\frac{5\pi}{4}]},\;{f^{-1}}(x)=\frac{\pi}{4}+{arccos}{\frac{x}{\sqrt{2}}}.")
CATEGORII :
- 1. BREVIAR TEORETIC pentru GIMNAZIU.
- 2. ALGORITMI IN MATEMATICA DE GIMNAZIU
-
3. BREVIAR TEORETIC pentru LICEU.
- 3.1. ELEMENTE DE LOGICA MATEMATICA (3)
- 3.2. MULTIMI NUMERICE (4)
- 3.3. NUMERE REALE (6)
- 3.4. IDENTITATI REMARCABILE (4)
- 3.5. INEGALITATI (5)
- 3.6. INECUATII (5)
- 3.7. ECUATII ALGEBRICE (6)
- 3.8. ECUATII TRANSCENDENTE (5)
- 3.9. NUMERE COMPLEXE (5)
- 3.10. PROGRESII (4)
- 3.11. COMBINATORICA (6)
- 3.12. LOGARITMI (6)
- 3.13. PROBABILITATI (4)
- 3.14. PERMUTARI (4)
- 3.15. DETERMINANTI (4)
- 3.16. MATRICE (5)
- 3.17. SISTEME DE ECUATII LINIARE (6)
- 3.18. SISTEME DE ECUATII NELINIARE (6)
- 3.19. CLASE DE RESTURI modulo n (4)
- 3.20. GRUPURI (4)
- 3.21. INELE SI CORPURI (4)
- 3.22. POLINOAME CU COEFICIENTI REALI (5)
- 3.23. POLINOAME CU COEFICIENTI COMPLECSI (4)
- 3.24. RELATII (4)
- 3.25. FUNCTII - generalitati (6)
- 3.26. FUNCTII ELEMENTARE (5)
- 3.27. FUNCTII SPECIALE (5)
- 3.28. FUNCTII INVERSABILE (5)
- 3.29. GRAFICE DE FUNCTII ELEMENTARE (12)
- 3.30. LIMITE DE SIRURI (4)
- 3.31. LIMITE DE FUNCTII (4)
- 3.32. FUNCTII CONTINUE (4)
- 3.33. FUNCTII DERIVABILE (4)
- 3.34. PROPRIETATI ALE FUNCTIILOR DERIVABILE (4)
- 3.35. PRIMITIVE (5)
- 3.36. INTEGRALE DEFINITE (6)
- 3.37. SCHIMBARI DE VARIABILA (6)
- 3.38. APLICATII ALE INTEGRALEI DEFINITE (4)
- 3.39. VECTORI (7)
- 3.40. TRIGONOMETRIE (7)
- 3.41. APLICATII ALE TRIGONOMETRIEI IN GEOMETRIE (4)
- 3.42. GEOMETRIE SINTETICA IN PLAN (8)
- 3.43. GEOMETRIE SINTETICA IN SPATIU (6)
- 3.44. GEOMETRIE ANALITICA IN PLAN (12)
- 3.45. GEOMETRIE ANALITICA IN SPATIU (4)
- 4. ALGORITMI IN MATEMATICA DE LICEU
- 5. CUM ABORDAM O PROBLEMA? (0)
- 6. PROBLEME DIVERSE CU REZOLVARI COMPLETE-LICEU (26)
- 7. REZOLVARI ELEMENTARE SI NEELEMENTARE - LICEU (8)
- 8. ALGEBRA-aplicatii-LICEU
- 9. PROBABILITATI-aplicatii-LICEU (10)
- 10. GEOMETRIE-aplicatii-LICEU
- 11. TRIGONOMETRIE-aplicatii-LICEU (33)
- 12. ANALIZA-aplicatii-LICEU
- 13. AUDITII-rezolvari-LICEU (4)
- 14. CUVINTE DE SPIRIT DESPRE MATEMATICA (0)
- 15. PROBLEME DISTRACTIVE (8)
- 16. UNDE ESTE GRESEALA ?