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Question Number 185574 by mnjuly1970 last updated on 23/Jan/23

     f (x )= cos(2πx)+ sin(2πx) +(√( ⌊x⌋ +⌊−x ⌋))             find        R_( f)  =?

$$ \\ $$$$\:\:\:{f}\:\left({x}\:\right)=\:{cos}\left(\mathrm{2}\pi{x}\right)+\:{sin}\left(\mathrm{2}\pi{x}\right)\:+\sqrt{\:\lfloor{x}\rfloor\:+\lfloor−{x}\:\rfloor} \\ $$$$\:\:\: \\ $$$$\:\:\:\:\:\:{find}\:\:\:\:\:\:\:\:{R}_{\:{f}} \:=? \\ $$

Commented by mahdipoor last updated on 23/Jan/23

(√([x]+[−x]))= { ((0                  x∈Z)),(((√(−1))=i     x∉Z)) :}    ⇒D_f =Z⇒f(x)=cos(2πk)+sin(2πk)+0=1

$$\sqrt{\left[{x}\right]+\left[−{x}\right]}=\begin{cases}{\mathrm{0}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{x}\in{Z}}\\{\sqrt{−\mathrm{1}}={i}\:\:\:\:\:{x}\notin{Z}}\end{cases}\:\: \\ $$$$\Rightarrow{D}_{{f}} ={Z}\Rightarrow{f}\left({x}\right)={cos}\left(\mathrm{2}\pi{k}\right)+{sin}\left(\mathrm{2}\pi{k}\right)+\mathrm{0}=\mathrm{1} \\ $$

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