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Data publicarii: 18 Decembrie, 2010

FUNCTII ELEMENTARE INVERSABILE

  • Functia de gradul intai, inversabila pe R:

Definitii:

f:R - > R, f(x) = y = ax + b,

unde a, b sunt numere reale, a nenul.

Inversa (tot functie de gradul intai):

f^{-1}:R\rightarrow{R}, f^{-1}(y)=x=\frac{y-b}{a}.f^{-1}:R\rightarrow{R}, f^{-1}(y)=x=\frac{y-b}{a}.

  • Functia de gradul al doilea, inversabila, separat, pe intervalele:

I1 = (-oo,-b/2a] si I2 = [-b/2a,+oo):

Definitii:

1) f:(-oo,-b/2a] - > (-oo,-Δ/4a],

f(x) = y = ax² + bx + c,

unde a < 0, b,c ЄR.

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}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:(-oo,-b/2a] - > (-Δ/4a,+oo),

f(x) = y = ax² + bx + c,

unde a > 0, b,cЄR.

Inversa:

f^{-1}:{[-\frac{\Delta}{4a},+\infty)}\rightarrow{(-\infty,-\frac{b}{2a},]},\;{f^{-1}}(y)=x=\frac{-b-\sqrt{b^2-4ac+4ay}}{2a}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,+oo) - > (-oo,-Δ/4a],

f(x) = y = ax² + bx + c,

unde a < 0, b,cЄR.

Inversa:

f^{-1}:{(-\infty,-\frac{\Delta}{4a}]}\rightarrow{[-\frac{b}{2a},+\infty)},\;{f^{-1}}(y)=x=\frac{-b+\sqrt{b^2-4ac+4ay}}{2a}.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,+oo) - > [-Δ/4a,+oo),

f(x) = y = ax² + bx + c,

unde a > 0, b,cЄR.

Inversa:

f^{-1}:{[-\frac{\Delta}{4a},+\infty)}\rightarrow{[-\frac{b}{2a},+\infty)},\;{f^{-1}}(y)=x=\frac{-b+\sqrt{b^2-4ac+4ay}}{2a}.f^{-1}:{[-\frac{\Delta}{4a},+\infty)}\rightarrow{[-\frac{b}{2a},+\infty)},\;{f^{-1}}(y)=x=\frac{-b+\sqrt{b^2-4ac+4ay}}{2a}.

Functia putere cu exponent natural impar, mai mare sau egal cu 3, 

inversabila pe R.

Definitie:

f:R - > R,

f(x)=y=x^{2n+1},f(x)=y=x^{2n+1},

unde nЄN*.

Inversa (functia radical de ordin impar):

f^{-1}:{\mathbb{R}}\rightarrow{\mathbb{R}},\;f^{-1}(y)=x=\sqrt[2n+1]{y}.f^{-1}:{\mathbb{R}}\rightarrow{\mathbb{R}},\;f^{-1}(y)=x=\sqrt[2n+1]{y}.

  • Functia putere cu exponent natural par, mai mare sau egal cu 2, inversabila, separat, pe intervalele

I1 = (-oo,0] si I2 = [0,+oo).

Definitii:

1) f:(-oo,0] - > [0,+oo),

f(x)=y=x^{2n},f(x)=y=x^{2n},

unde nЄN*.

Inversa (opusa functiei radical de ordin par):

f^{-1}:{[0,+\infty)}\rightarrow{(-\infty,0]},\;f^{-1}(y)=x=-\sqrt[2n]{y}.f^{-1}:{[0,+\infty)}\rightarrow{(-\infty,0]},\;f^{-1}(y)=x=-\sqrt[2n]{y}.

2) f:[0,+oo) - > [0,+oo),

f(x)=y=x^{2n},f(x)=y=x^{2n},

unde nЄN*.

Inversa:

f^{-1}:{[0,+\infty)}\rightarrow{([0,+\infty)},\;f^{-1}(y)=x=\sqrt[2n]{y}.f^{-1}:{[0,+\infty)}\rightarrow{([0,+\infty)},\;f^{-1}(y)=x=\sqrt[2n]{y}.

  • Functia putere, cu exponent real nenul, este inversabila pe multimea numerelor reale pozitive.

Definitie:

f:(0,+oo) - > (0,+oo),

f(x)=y=x^{\alpha},f(x)=y=x^{\alpha},

oricare ar fi αЄR*.

Inversa:

{f^{-1}}:{(0+\infty)}\rightarrow{(0,+\infty)},{f^{-1}}(y)=x={y}^{\frac{1}{\alpha}}.{f^{-1}}:{(0+\infty)}\rightarrow{(0,+\infty)},{f^{-1}}(y)=x={y}^{\frac{1}{\alpha}}.

  • Functia exponentiala, inversabila pe R.

Definitie:

f:R - > (0,+oo),

f(x)=y=a^x,f(x)=y=a^x,

unde aЄ(0,1)U(1,+oo);

Inversa (functia logaritmica):

f^{-1}:{(0,+\infty)}\rightarrow{\mathbb{R}},\;{f^{-1}}(y)=x={log}_a{y}.f^{-1}:{(0,+\infty)}\rightarrow{\mathbb{R}},\;{f^{-1}}(y)=x={log}_a{y}.

Deci:

y=a^xy=a^x <=> x=\log_ay,x=\log_ay,

unde xЄR, yЄ(0,+oo), iar aЄ(0,1)U(1,+oo).

  • Functia sinus, restrictionata la intervalul [-π/2,π/2], anume

f:[-π/2;π/2] - > [-1;1],

f(x) = y = sinx  

este inversabila si admite:

Inversa (arcsinus):

f^{-1}:{[-1,+1]}\rightarrow{[-\frac{\pi}{2},+\frac{\pi}{2}]},\;{f^{-1}}(y)=x=arcsiny.f^{-1}:{[-1,+1]}\rightarrow{[-\frac{\pi}{2},+\frac{\pi}{2}]},\;{f^{-1}}(y)=x=arcsiny.   

Deci:

y = sinx <=> x = arcsiny, unde xЄ[-π/2;π/2], iar yЄ[-1;1].

  • Functia cosinus, restrictionata la intervalul [0,π], anume

f:[0,π] - > [-1;1], f(x) = y = cosx, este inversabila si admite:

Inversa (arccosinus):                                     

{f^{-1}}:[-1;1]\rightarrow{[0;\pi]},{f^{-1}}(y)=x={arccosy}.{f^{-1}}:[-1;1]\rightarrow{[0;\pi]},{f^{-1}}(y)=x={arccosy}.

Deci:

y = cosx <=> x = arccosy,

unde xЄ[o,π], iar yЄ[-1;1].

  • Functia tangenta, restrictionata la intervalul (-π/2,π/2), anume

f:(-π/2,π/2) - > R,

f(x) = y = tgx, este inversabila si admite:

Inversa (arctangenta):

{f^{-1}}:{\mathbb{R}}\rightarrow{(-\frac{\pi}{2};\frac{\pi}{2})},{f^{-1}}(x)=y=arctgx.{f^{-1}}:{\mathbb{R}}\rightarrow{(-\frac{\pi}{2};\frac{\pi}{2})},{f^{-1}}(x)=y=arctgx.

Deci:

y = tgx <=> x = arctgy,

unde xЄ(-π/2,π/2), iar yЄR.

  • Functia cotangenta, restrictionata la intervalul (0,π), anume

f:(0,π) - > R,

f(x) = y = ctgx, este inversabila si admite:

Inversa (arccotangenta):

{f^{-1}}:{\mathbb{R}}\rightarrow{(0,\pi)},\;{f^{-1}}(y)=x={arcctgx}.{f^{-1}}:{\mathbb{R}}\rightarrow{(0,\pi)},\;{f^{-1}}(y)=x={arcctgx}.

Deci:

y = ctgx <=> x = arcctgy,

unde xЄ(0,π), iar yЄR.                  

 


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Răspunsuri şi comentarii

tema

Alin, 14.07.2016 11:30

EU nu inteleg ceva aici voi aratati doar inversa ei dar cum aratam ca acea functie este inversabila??

Răspuns: Pentru a intelege TOATE detaliile, e necesara cunoasterea teoriei. In acest caz, este vorba de definitii cum ar fi pentru functii injective, surjective, bijective, In plus, e necesara cunoasterea tuturor functiilor elementare (definitii, proprietati...). Asta e !

comentariu

Gabriela, 14.03.2011 22:48

Foarte bine realizata sinteza de aici! Are suficienta informatie si, in acelasi timp, este si usor de inteles.

Răspuns: Mul?umesc pentru aprecieri!

 

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