Question

Daca functia liniara f:R->R, f(x)=ax+, verifica egaalitatea: f(x+1)=f²(1)+[(2x+1)/4], oricare ar fi x∈R, atunci ab este?

Answer 1

Răspuns:

f(x) = ax +b, f(x+1) = f²(1) + [(2x+1)/4]

f(x+1) = a(x+1) + b

f(x+1) = f²(1) + [(2x+1)/4]  =>

a(x+1) + b = f²(1) + [(2x+1)/4]  

dar f(1) = a*1+b => f²(1) = (a+b)² =>

a(x+1) + b = (a+b)² + [(2x+1)/4]

ax + a - (a+b)² = 2x/4 +  1/4  <=>

a = 2/4   si    a - (a+b)² = 1/4 =>

a = 1/2 si 1/2 - (1/2 +b)² = 1/4 =>

(1/2 +b)² = 1/2 - 1/4

1/4 + b + b² = 1/4

b²+b = 0

b(b+1) = 0 => b1 = 0 , b2 = -1,  daca b≠0 ramane b = -1

a = 1/2 , b = -1

f(x) = x/2 -1