Question

SA SE ARATE CA FUNCTIILE SUNT BIJECTIVE SI SA SE AFLE INVERSELE ACESTORA:

a) f:(-∞, 1), (4+∞), f(x)=5-x

b) f:(-∞,2) ,(-∞,10), f(x)=3x+4

Answer 1

a) injectivitate : x1/= x2 => f(x1)/= f(x2) 5-x1 /= 5-x2 => -x1/=-x2 => x1/=x2, Adevarat => f este injectiva Surjectivitate : oricare ar fi y apartine (4, infinit) exista x apartine (- infinit, 1) astfel incat f(x)= y => 5-x =y => x=5-y y>4 => x<1, adevarat => f surjectiva => f bijectiva f ^-1 (x) =5-x b)x1/= x2 => f(x1)/=f(x2) => 3x1+4 /=3x2+4 => 3x1/=3x2 => x1/=x2 Adeverat => f injectiva Surjectivitate : oricare ar fi y apartine (- infinit, 10) exista x apartine (- infinit, 2) astfel incat f(x)= y 3x+4 =y => 3x=y-4 => x= (y-4)/3 y<10 => y-4<6 => x<2 Adevarat => f surjectiva deci bijectiva f^-1(x)=(x-4)/3