Question

Consider the real-valued function satisfying $2f(sinx)+f(cosx)=x$. Then, which of the following is not true?

• A. Domain of $f(x)$ is $[-1$, $1]$
• B. Range of $f(x)$ is $\left[-\frac{2\pi }{3},\frac{\pi }{3}\right]$
• C. $f(x)$ is one-one
• D. None of these

Answer 1

Answer: (D)

Given, $2f(sinx)+f(cosx)=x\dots (i)$
Replace $x$ by $\frac{\pi }{2}-x$, we have
$2f\left(cosx\right)+f\left(sinx\right)=\frac{\pi }{2}-x\dots \left(ii\right)$
Eliminating $f(cosx)$ from $(i)$ and $(ii)$, we have
$f\left(sinx\right)=x-\frac{\pi }{6}$
$\Rightarrow f\left(x\right)=si{n}^{-1}x-\frac{\pi }{6}$
Then, domain of $f(x)$ is $[-1$, $1]$
Range is $\left[-\frac{\pi }{2}-\frac{\pi }{6},\frac{\pi }{2}-\frac{\pi }{6}\right]$ or $\left[-\frac{2\pi }{3},\frac{\pi }{3}\right]$
Also, $f(x)$ is one-one.