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  4. If f(x)= beginBmatrix[x], text if -3< x ≤-1 |x|, text if -1< x< 1 |[x]|, text if 1 ≤ x< 3 endBmatrix the set of values of x, where f(x) ≥ 0, (where [.] represents the greatest integer function) is

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Q. If $f(x)=\begin{Bmatrix}{[x], \text { if }-3< x \leq-1} \\ |x|, \text { if }-1< x< 1 \\ |[x]|, \text { if } 1 \leq x< 3\end{Bmatrix}$
the set of values of $x$, where $f(x) \geq 0$, (where [.] represents the greatest integer function) is

NTA AbhyasNTA Abhyas 2022Relations and Functions - Part 2

Solution:

Given,
$f x=x$ if $-3< x \leq-1 x$ if $-1< x< 1 x$ if $1 \leq x< 3$
When $-3< x \leq-1, f x=x \Rightarrow f x<0$
When $-1< x< 1 , f x=x \Rightarrow f x \geq 0$
When $1 \leq x< 3, f x=x \Rightarrow f x \geq 1$
$\therefore$ The set $x: f x \geq 0=(-1,3)$.