Question

If function $\mathrm{f}\left(\mathrm{x}\right)=\frac{1}{2}-\tan \left(\frac{\mathrm{\pi x}}{2}\right);(-1<\mathrm{x}<1)$ and $\mathrm{g}\left(\mathrm{x}\right)=\sqrt{3+4\mathrm{x}-4{\mathrm{x}}^2}$, then domain of $(f+g)$ is given by

• A. $\left(\frac{1}{2},1\right)$
• B. $\left(\frac{1}{2},-1\right)$
• C. $\left(\frac{-1}{2},1\right)$
• D. $\left(\frac{-1}{2},-1\right)$

Answer 1

The correct option is C

$\left(\frac{-1}{2},1\right)$

Explanation for the correct option:

Step 1: Given information

$\mathrm{f}\left(\mathrm{x}\right)=\frac{1}{2}-\tan \left(\frac{\mathrm{\pi x}}{2}\right);\left(-1<\mathrm{x}<1\right)$

$\mathrm{g}\left(\mathrm{x}\right)=\sqrt{3+4\mathrm{x}-4{\mathrm{x}}^2}$

Step 2: Finding the domain of $(f+g)$

Domain of $(f+g)=D\left(F\right)\cap D\left(G\right)$

$$\begin{gathered} 3+4\mathrm{x}-4{\mathrm{x}}^2\ge 0 \\ 4{\mathrm{x}}^2-4\mathrm{x}-3\le 0 \\ (2\mathrm{x}+1)(2\mathrm{x}-3)\le 0 \\ \mathrm{x}\in \left[-\frac{1}{2},1\right] \end{gathered}$$

Hence, the correct option is option (c).