Question

If $f(x)=\sin ^{-1}\left(\frac{4 x}{4+x^2}\right)-\frac{2}{3} \tan ^{-1}\left(\frac{x}{2}\right)$ and $g(x)=\begin{cases}2|x|-2, & 1 \leq|x| \leq 2 \\ \ln (1+[|x|]), & -1< x< 1\end{cases}$
where $[ k ]$ denotes greatest integer less than or equal to $k$, and $x \in[-2,2]$, then number of solution(s) of the equation $| f ( x )|= g ( x )$ is(are)

• A. 4
• B. 3
• C. 2
• D. 1

Answer 1

Answer: (B)

$f(x)=\sin ^{-1}\left(\frac{4 x}{4+x^2}\right)-\frac{2}{3} \tan ^{-1} \frac{x}{2}$
$=\sin ^{-1}\left(\frac{2 \cdot\left(\frac{x}{2}\right)}{1+\left(\frac{x}{2}\right)^2}\right)-\frac{2}{3} \tan ^{-1} \frac{x}{2}$
For $x \in[-2,2]$
$f(x)=2 \tan ^{-1} \frac{x}{2}-\frac{2}{3} \tan ^{-1} \frac{x}{2}=\frac{4}{3} \tan ^{-1} \frac{x}{2}$