Functions Question 699
Question: The number of linear functions f satisfying $ f(x+f(x))=x+f(x)\forall x\in R $ is
Options:
A) 0
B) 1
C) 2
D) 3
Show Answer
Answer:
Correct Answer: C
Solution:
[c] Let $ f(x)=ax+b $ (1)
$ \Rightarrow f(ax+b+x)=x+ax+b $
$ \Rightarrow f((a+1)x+b)=(a+1)x+b $ Replace $ (a+1)x+b $ by, y, we have
$ \Rightarrow f(y)=(a+1)( \frac{y-b}{a+1} )+b $ Or $ f(x)=(a+1)( \frac{x-b}{x+1} )+b $
$ \therefore $ Required no, of linear functions is 2.
BETA
