Question

Sa se determine functia g:R --> R, daca:

a) (gof)(x)=4x-3, unde f:R->R, f(x)=x+3;
b) (fog)(x)=-3x+4, unde f:R->R,f(x)=x+1;

Answer 1

a)

g(x)=ax+b

f(x)=x+3

(g°f)(x)=g(f(x))=g(x+3)=a(x+3)+b=ax+3a+b

(g°f)(x)=4x-3

ax+3a+b=4x-3

{a=4

{3a+b=-3

3*4+b=-3

12+b=-3

b=-15

g:R->R, g(x)=4x-15

b)

g(x)=ax+b

f(x)=x+1

(f°g)(x)=f(g(x))=f(ax+b)=ax+b+1

(f°g)(x)=-3x+4

ax+b+1=-3x+4

{a=-3

{b+1=4 => b=3

g: R->R, g(x)=-3x+3

PS: Ne dam seama ca functia g este functie de gradul I, pentru ca atat compunerea functiilor f si g, cat si functia f sunt functii de gradul I.