Răspuns:
Explicație pas cu pas:
f(x)=ax+b. Tr. gasiti a si b.
a) f(x+1)=2x+3, ⇒a(x+1)+b=2x+3, ⇒ax+a+b=2x+3, deci a=2, a+b=3, 2+b=3, b=3-2=1. deci f(x)=2x+1.
b) f(5-2x)=2x-4, ⇒a(5-2x)+b=2x-4, ⇒-2ax+5a+b=2x-4, ⇒-2a=2, de unde a=-1. Si 5a+b=-4, ⇒5·(-1)+b=-4, ⇒-5+b=-4, ⇒b=-4+5=1. Deci f(x)=-x+1.
c) 3·f(x+1)-4=2x, ⇒3·(a(x+1)+b)=2x+4, ⇒3ax+3+3b=2x+4, ⇒3a=2, deci a=2/3
3+3b=4, ⇒3b=4-3, ⇒3b=1, ⇒b=1/3. Deci f(x)=(2/3)·x + (1/3).
d) f(x)+f(x+1)=2x+1, ⇒ax+b+a(x+1)+b=2x+1, ⇒ax+b+ax+a+b=2x+1, ⇒2ax+a+2b=2x+1, ⇒2a=2, deci a=1. Si a+2b=1, ⇒1+2b=1, ⇒2b=0, ⇒b=0.
Deci f(x)=1·x+0, adica f(x)=x.
e) 2·f(x-1)-f(3x)=4-x, ⇒2·(a(x-1)+b)-a·3x+b=4-x, ⇒2ax-2a+2b-3ax+b=-1·x+4, ⇒5ax-2a+3b=-1·x+4, ⇒5a=-1, deci a=-1/5. Si -2a+3b=4, ⇒2/5 +3b=4, ⇒3b=4- 2/5, ⇒3b=20/5 - 2/5, ⇒3b=18/5, ⇒b=(18/5):3=6/5.
Deci f(x)=-(1/5)x +(6/5).
