Explanation

(a)
Given,
$f(x) = \left\{ \begin{array}{r}
\lbrack x\rbrack\ \ if\ - 3 < x \leq - 1 \\
|x|\ \ \ \ if - 1 < x < 1 \\
\left| \lbrack x\rbrack \right|\ \ \ \ if\ \ 1 \leq x < 3
\end{array} \right.\$
When $- 3 < x \leq - 1,\ \ \ \ f(x) = \lbrack x\rbrack\ \ \ \Rightarrow \ \ \ f(x) < 0$
When $- 1 < x < 1,\ \ \ \ \ \ f(x) = |x|\ \ \ \Rightarrow \ \ \ \ f(x) > 0$
When $1 \leq x < 3,\ \ f(x) = \left| \lbrack x\rbrack \right|\ \Rightarrow \ \ \ \ f(x) > 0$
$\therefore$ The set $\left( x\ :f(x) \geq 0 \right) = ( - 1,\ 3)$.