What is the range of f(x) = [[x]/x] Question is of Greatest Integer Function 

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Please find below the solution to the asked query:

$$\begin{gathered} \mathrm{We} \mathrm{have} \\ \mathrm{f}\left(\mathrm{x}\right)=\left[\frac{\left[\mathrm{x}\right]}{\mathrm{x}}\right] \\ \mathrm{Function} \mathrm{is} \mathrm{not} \mathrm{defined} \mathrm{for} \mathrm{x}=0 \\ \mathrm{We} \mathrm{know} \mathrm{that} \\ \left[\mathrm{x}\right]\le \mathrm{x} \\ \mathrm{When} \mathrm{x}>0 \\ \Rightarrow \frac{\left[\mathrm{x}\right]}{\mathrm{x}}\le 1 \\ \mathrm{and} \mathrm{when} \mathrm{x}<0 \\ \frac{\left[\mathrm{x}\right]}{\mathrm{x}}\ge 0 \\ \mathrm{Hence} \frac{\left[\mathrm{x}\right]}{\mathrm{x}} \mathrm{will} \mathrm{achieve} \mathrm{each} \mathrm{non}-\mathrm{negative} \mathrm{integer}, \mathrm{hence} \\ \mathrm{Range} \mathrm{of} \left[\frac{\left[\mathrm{x}\right]}{\mathrm{x}}\right] \mathrm{is} \left\{0,1,2,3,....\right\} \end{gathered}$$

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