Question

Domain of function $y=\sqrt{{\log }_{0.6}\left(\frac{x-1}{x+5}\right)}\cdot \frac{1}{(x^2-49)}$ is?

• A. $(1,7)\cup (7,\infty )$
• B. $R-(1,7)$
• C. $R-\left\{7\right\}$
• D. $(-7,7)$

Answer 1

The correct option is A $(1,7)\cup (7,\infty )$

$\begin{array}{c}\text{Solution - Given that,}\\ \begin{array}{c}y=\sqrt{{\log }_{0.6}\left(\frac{x-1}{x+5}\right)}\cdot \frac{1}{(x^2-49)}\\ x^2-49\ne 0\\ (x-7)(x+7)\ne 0\\ x\ne 7,-7\end{array}\\ \text{Now,}\\ {\log }_{0.6}\left(\frac{x-1}{x+5}\right)⩾0\end{array}$

$\begin{array}{cc}\Rightarrow & \frac{x-1}{x+5}⩽(0,6{)}^0\\ \Rightarrow \frac{x-1}{x+5}⩽1& \Rightarrow \frac{x-1}{x+5}-1⩽0\\ \Rightarrow \frac{x-1-x-5}{x+5}⩽0& \\ \Rightarrow +\frac{6}{x+5}⩾0& \Rightarrow x+5>0\\ \text{Also, for log to be defined,}\frac{x-1}{x+5}>0& \end{array}$ $\begin{array}{c}x\in (1,7)\cup (7,\infty )\\ \text{Hence,}\left(A\right)\text{is the correct option.}\end{array}$