a)surjectivitatea
x->-∞, f(x)->∞; x->1, f(x) ->5-1=4
injectivitatea ; x1≠x2, x1-x2≠0⇒
f(x1)-f(x2)=5-x1- (5-x2)=x2-x1≠0 deci f(x1)≠f(x2)
deci bijectiva
y=-x+5
x=y+5
f^(-1)(x)= x+5, functia inversa
b) x->-∞, f(x)->-∞
x=2, y=3*2+4=10
surjectiva
x1≠x2, f(x1)-f(x2) =3(x1-x2)≠0, f(x1)≠f(x2) injectiova
deci bijectiva
functia inversa:
y=3x+4
3x=y-4
x=y/3 -4/3
f^(-1) (x)=x/3-4/3
c) x->-2, x>-2, f(x)->-7/+0=-∞
x->-2, x<-2 f(x)->-7/-0=+∞
x->∞, f(x) ->3; x->-∞, f(x) ->3x/x=3
f(x) =(3x-1)/(x+2)=(3x+6-7)/(x+2)=3-7/(x+2)
x+2 crescatoare pe R\{-2}, 1/(x+2) descrescatoare,, -7/(x+2) crescatoare, 3-7/(x+2) crescatoare pe R\{-2} deci injectiva cum ia toate valorile din R\{3} f(x) surjectiva si injectiva, deci bijectiva
functia inversa:
y=(3x-1)/(x+2)
xy+2y=3x-1
xy-3x=-1-2y
3x-xy=2y+1
x( 3-y)=2y+1
x= (2y+1)/(3-y)
f^(-1)(x)= (2x+1)/(3-x)
d) analog
(6x+1)/(3x-4
)
la +4/3 ->∞
la-4/3->-∞
la ∞si la -∞ tinde cate 6x/2x=2
(6x-8+9)/(3x-4)=2+9/(3x-4)
3x-4 crescatoare, 9/(3x-4) descrescatoare, 9/(3x-4) descrescatoare pr R\ {4/3} deci injectiva cum ia toate valorile intre -∞si ∞ este si surjectiva, deci bijectiva
Functia inversa:
f(x) =y=(6x+1)/(3x-4)
3xy-4y=6x+1
3xy-6x=4y+1
x(3y-6)=4y+1
x=(4y+1)/(3y-6)
f^(-1)(x)= (4x+1)/(3x-6)
Bonus ; grafice
f(x) = (3x-1)/(x+2)
si
f(x)= (6x+1)/(3x-4)
