Question

Fie f, g: R->R care satisfac (f o g)(x)=2x+1
a)determinati functia f daca g(x)=3x-2
b)determinati functia g daca f(x)=4x+1

Answer 1

Daca presupunem ca f si g sunt functii de gradul I adica ax+b, avem: 
a) f(x)= ax+b , si g(x)=3x-2 ⇒ (fog) = ag+b= a(3x-2) +b =3ax-2a+b =2x+1 (din ipoteza) deci identificand coeficientii avem: 3a=2 si -2a+b=1 ⇒ a=2/3 si b=7/3 ⇒ f(x)=(2/3)x+7/3
b) g(x)=ax+b, si f(x)=4x+1, ⇒ (fog)(x)=f(g(x))=4g(x)+1=4(ax+b)+1=4ax+4b+1=2x+1
( din ipoteza) , ⇒ 4a=2 si 4b+1=1 ⇒ a=1/2 si b=0, g(x)=(1/2)x.