Write the range of the fuction f(x)=e^x-[x] ,x E R

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$$\begin{gathered} \mathrm{We} \mathrm{have}: \\ \mathrm{f}\left(\mathrm{x}\right)={\mathrm{e}}^{\mathrm{x}-\left[\mathrm{x}\right]} \\ \mathrm{We} \mathrm{know} \mathrm{that} \\ \mathrm{x}-\left[\mathrm{x}\right]=\left\{\mathrm{x}\right\}, \mathrm{where} \left\{ \right\} \mathrm{represents} \mathrm{fractional} \mathrm{part} \mathrm{of} \mathrm{x}. \\ \Rightarrow \mathrm{f}\left(\mathrm{x}\right)={\mathrm{e}}^{\left\{\mathrm{x}\right\}} \\ \mathrm{Now} \\ 0\le \left\{\mathrm{x}\right\}<1 \\ \Rightarrow {\mathrm{e}}^0\le {\mathrm{e}}^{\left\{\mathrm{x}\right\}}<{\mathrm{e}}^1 \\ \Rightarrow 1\le {\mathrm{e}}^{\left\{\mathrm{x}\right\}}<{\mathrm{e}}^1 \\ \mathbf{Hence}\mathbf{ }\mathbf{range}\mathbf{ }\mathbf{of}\mathbf{ }\mathbf{f}\left(\mathbf{x}\right)\mathbf{=}{\mathbf{e}}^{\mathbf{x}\mathbf{-}\left[\mathbf{x}\right]}\mathbf{ }\mathbf{is}\mathbf{ }\mathbf{[}\mathbf{1}\mathbf{,}\mathbf{ }\mathbf{e}\mathbf{)} \end{gathered}$$

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