Question

If f (f(x)) = x + 1 for all x ∈ R and if $f\left(0\right)=\frac{1}{2},$ then f(1) = ____________.

Answer 1

Given: f (f(x)) = x + 1 for all x ∈ R and $f\left(0\right)=\frac{1}{2}$

$$\begin{gathered} f\left(f\left(x\right)\right)=x+1 \\ \Rightarrow f\left(f\left(0\right)\right)=0+1 \\ \Rightarrow f\left(\frac{1}{2}\right)=1\left(\because f\left(0\right)=\frac{1}{2}\right)....\left(1\right) \\ \mathrm{Now}, \\ f\left(f\left(\frac{1}{2}\right)\right)=\frac{1}{2}+1 \\ \Rightarrow f\left(1\right)=\frac{1+2}{2}\left(\because f\left(\frac{1}{2}\right)=1\right) \\ \Rightarrow f\left(1\right)=\frac{3}{2} \end{gathered}$$

Hence, f(1) = $\underline{\frac{3}{2}}$.