Question

If a function $ f $ satisfies $ f\{f(x)\}=x+1 $ for all real values of x and if $ f(0)=\frac{1}{2}, $ then $ f(1) $ is equal to

• A. $ \frac{1}{2} $
• B. 1
• C. $ \frac{3}{2} $
• D. 2
• E. 0

Answer 1

Answer: (C)

Given, $ f\{f(x)\}=x+1 $ ...(i)
$ \therefore $ $ f(f(0))=0+1 $
$ \Rightarrow $ $ f\left( \frac{1}{2} \right)=1 $
$ \left[ \because f(0)=\frac{1}{2} \right] $
Now, put $ x=\frac{1}{2} $ in Eq. (i), we get
$ f\left\{ f\left( \frac{1}{2} \right) \right\}=\frac{1}{2}+1 $
$ \Rightarrow $ $ f(1)=\frac{3}{2} $