Question

The function is $f(x)=\frac{1}{2-\cos 3x},x\in \left[0,\frac{\pi }{3}\right],$ is

• A. one-one, but not onto
• B. onto, but not one-one
• C. one-one as well as onto
• D. neither one-one nor onto

Answer 1

Answer: (C)

Given, $f(x)=\frac{1}{2-\cos 3x},x\in \left[0,\frac{\pi }{3}\right]$ For one - one Let $f(x_1)=f(x_2)$
$\Rightarrow$ $\frac{1}{2-\cos 3x_1}=\frac{1}{2-\cos 3x_2}$
$\Rightarrow$ $2-\cos 3x_1=2-\cos 3x_2$
$\Rightarrow$ $\cos 3x_1=\cos 3x_2\Rightarrow x_1=x_2$
$\Rightarrow$ f is one-one For onto Let $y=f(x),y\in$ codomain
$\Rightarrow$ $y=\frac{1}{2-\cos 3x}$
$\Rightarrow$ $y(2-\cos 3x)=1$
$\Rightarrow$ $2-\cos 3x=\frac{1}{y}$
$\Rightarrow$ $\cos 3x=2-\frac{1}{y}$
$\Rightarrow$ $x=\frac{1}{3}{\cos }^{-1}\left(2-\frac{1}{y}\right)$
Here, for all $y\in$ codomain there exist $x\in$ domain. so $f(x)$ is onto.
Here for all $yE$ codomain there exist $x\in$ domain. so is on to. $f(x)$