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Question Number 205297 by mnjuly1970 last updated on 15/Mar/24

$$$$$$Find{the}^{″}range{}^{″}of:$$$$$$$$i:f\left(x\right)=\lfloor \frac{x}{\lfloor x\rfloor }\rfloor$$$$ii:f\left(x\right)=\frac{x}{\lfloor x\rfloor +\lfloor -x\rfloor }$$$$$$

Answered by A5T last updated on 15/Mar/24

$$\left(i\right)fisundefinedwhen\lfloor x\rfloor =0(0⩽x<1)$$$$Otherwisex=\lfloor x\rfloor +\left\{x\right\},where0⩽\left\{x\right\}<1$$$$\Rightarrow f\left(x\right)=\lfloor 1+\frac{\left\{x\right\}}{\lfloor x\rfloor }\rfloor ,whenx⩾1,\left\{x\right\}<\lfloor x\rfloor ,\Rightarrow f\left(x\right)=1$$$$whenx<0f\left(x\right)=\lfloor 1-\frac{\left\{x\right\}}{\mid \lfloor x\rfloor \mid }\rfloor ;\left\{x\right\}<\mid \lfloor x\rfloor \mid$$$$\Rightarrow 0<1-\frac{\left\{x\right\}}{\mid \lfloor x\rfloor \mid }⩽1\Rightarrow f{\left(x\right)}_{x<0}=0or1$$$$\Rightarrow Range:\{0,1\}$$

Commented by mnjuly1970 last updated on 15/Mar/24

$$thanksalotali$$

Commented by Frix last updated on 15/Mar/24

$$\mathrm{Yes}.\mathrm{Just}\mathrm{to}\mathrm{clarify}:$$$$f\left(x\right)=\{\begin{array}{l}1;x\in {\mathbb{Z}}^{-}\\ 0;x\in {\mathbb{R}}^{-}\mathrm{\setminus }\mathbb{Z}\\ \mathrm{undefined};0⩽x<1\\ 1;x⩾1\end{array}$$

Answered by A5T last updated on 15/Mar/24

$$\left(ii\right)whenx\in \mathbb{Z},f\left(x\right)isundefined$$$$whenx>0,\lfloor -x\rfloor =-\lfloor x\rfloor -1\Rightarrow f\left(x\right)=-x$$$$whenx<0,\lfloor -x\rfloor =-\lfloor x\rfloor -1\Rightarrow f\left(x\right)=-x$$$$\Rightarrow f\left(x\right)\in \mathbb{R}\mathrm{\setminus }\mathbb{Z}$$

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