Resoluções de Análise Matemática, Demidovitch Cap. I Ex.20

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20. \( y=arcsen \left( lg\frac{x}{10} \right);\)

Resolução:

\( |lg\frac{x}{10}| \le 1\) \(\Longrightarrow\)
\(\left\{ \begin{array}{rll}
lg\frac{x}{10} \le 1, & \hbox{se} & lg\frac{x}{10} \ge 0, x \gt 0 \\
-lg\frac{x}{10} \le 1, & \hbox{se} & lg\frac{x}{10} \lt 0, x \gt 0
\end{array}\right.\)
\(\Longrightarrow\) \(\left\{ \begin{array}{rll}
\frac{x}{10} \le 10, & \hbox{se} & \frac{x}{10} \ge 1, x \gt 0 \\
\frac{x}{10} \ge 10^{-1}, & \hbox{se} & \frac{x}{10} \lt 1, x \gt 0
\end{array}\right.\)
\(\Longrightarrow\) \( \left\{ \begin{array}{rll}
x \le 100, & \hbox{se} & x \ge 10, x \gt 0 \\
x \ge 1, & \hbox{se} & x \lt 10, x \gt 0
\end{array}\right.\)

Entretanto, \( D_y=x:x\in [1 ; 100] \)