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Q.
Consider the real-valued function satisfying $2f(sinx) +f(cosx) = x$. Then, which of the following is not true?
Relations and Functions - Part 2
Solution:
Given, $2f(sinx) +f(cosx) = x \ldots (i)$
Replace $x$ by $\frac{\pi}{2}-x$, we have
$2f \left(cos\,x\right)+f \left(sin\,x\right)=\frac{\pi}{2}-x\ldots\left(ii\right)$
Eliminating $f(cosx)$ from $(i)$ and $(ii)$, we have
$f \left(sin\,x\right)=x-\frac{\pi}{6}$
$ \Rightarrow f \left(x\right)=sin^{-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.