Question

The domain of the function, $f ( x )=( x +0.5)^{\log _{(0.5+ x )}\left(\frac{ x ^2+2 x -3}{4 x ^2-4 x -3}\right)}$ is

• A. $\left(\frac{-1}{2}, \infty\right)$
• B. $[1,3]$
• C. $\left(\frac{1}{2}, 1\right) \cup\left(\frac{3}{2}, \infty\right)$
• D. $\left(\frac{-1}{2}, \frac{1}{2}\right) \cup\left(\frac{1}{2}, 1\right) \cup\left(\frac{3}{2}, \infty\right)$

Answer 1

Answer: (D)

$x+0.5>0 \Rightarrow x+0.5 \neq 1$
Also, $\frac{x^2+2 x-3}{4 x^2-4 x-3}>0 $
$\Rightarrow \frac{(x+3)(x-1)}{(2 x+1)(2 x-3)}>0$