Explanation

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