Question

Consider the function $f\left(x\right)=\sin \left({\sin }^{-1}2x\right)+\sec \left({\sec }^{-1}3x\right)+\tan \left({\tan }^{-1}4x\right)$ and $g\left(x\right)=9x,$ then which of the following is correct$?$

• A. $fog\left(x\right)$ and $gof\left(x\right)$ are equal function.
• B. $f\left(x\right)$ is an odd function
• C. Number of integers in range of $f\left(x\right)$ are $4.$
• D. Number of integers in domain of $f\left(x\right)$ are $2.$

Answer 1

Answer: (B), (C)

The correct options are
B $f\left(x\right)$ is an odd function
C Number of integers in range of $f\left(x\right)$ are $4.$

$\sin \left({\sin }^{-1}2x\right)=2x$ for $-\frac{1}{2}\le x\le \frac{1}{2}$
$\sec \left({\sec }^{-1}3x\right)=3x$ for $x\le -\frac{1}{3}$ or $x\ge \frac{1}{3}$
$\therefore$ $f\left(x\right)=9x$ for $xϵ[-\frac{1}{2},-\frac{1}{3}]\cup [\frac{1}{3},\frac{1}{2}]$
(A) $fog\left(x\right)=9g\left(x\right)=81x$ is defined for

$g\left(x\right)ϵ[-\frac{1}{2},-\frac{1}{3}]\cup [\frac{1}{3},\frac{1}{2}]$
while $gof\left(x\right)=9f\left(x\right)=81x$ is defined for
$xϵ[-\frac{1}{2},-\frac{1}{3}]\cup [\frac{1}{3},\frac{1}{2}]$
$\therefore$ $gof\left(x\right)$ & $fog\left(x\right)$ are not equal.
(C) Range of $f\left(x\right)=[-\frac{9}{2},-3]\cup [3,\frac{9}{2}]$
$\Rightarrow$ Number of integers$=4$