Question

If $f(x)={\sin }^{-1}\left(\frac{4x}{4+x^2}\right)-\frac{2}{3}{\tan }^{-1}\left(\frac{x}{2}\right)$ and $g(x)=\{\begin{array}{ll}2|x|-2,& 1\le |x|\le 2\\ \ln (1+[|x|]),& -1<x<1\end{array}$
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)

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