Functions Question 487
Question: The number of linear functions f satisfying $ f(x+f(x))=x+f(x)\forall x\in R $ is
Options:
0
1
2
3
Show Answer
Answer:
Correct Answer: C
Solution:
[c] Let $ f(x)=ax+b $ (1)
$ \Rightarrow f(ax+b+x)=ax+b+x $
$ \Rightarrow f((a+1)x+b)=(a+1)x+b $ Let $ (a+1)x+b $ be replaced by $ y $, we have
$ \Rightarrow f(y)=(a+1)( \frac{y-b}{a+1} )+b $ Or $ f(x)=(a+1)( \frac{x-b}{a} )+b $
$ \therefore $ Required number of linear functions is 2.
BETA
