Question

(25 PONTOS) Seja $A$ o domínio da função $\mathrm{tg\,}x,$ isto é,
$A=\left\{x \in \mathbb{R}\left|\,x \neq \dfrac{\pi}{2}+n\pi,\;n \in \mathbb{Z}\right. \right \}.$
Existe apenas um $k \in \mathbb{Z}$, tal que
para todo $x \in A,$
$\mathrm{arctg\,}(\mathrm{tg\,}x)=x+k\pi$

Determine em qual intervalo real está este inteiro $k.$
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Resposta: $k \in \mathbb{Z}\;\text{ e }\;-\dfrac{1}{2}-\dfrac{x}{\pi}\ \textless \ k\ \textless \ \dfrac{1}{2}-\dfrac{x}{\pi}.$

Answer 1

El rango de la función arcotangente es $\mbox{Ran}(\arctan)=\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)$

entonces
$x+\pi k \in\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)\iff -\dfrac{\pi}{2}\ \textless \ x+\pi k\ \textless \ \dfrac{\pi}{2}$