First slide

Functions (XII)

Question

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

Moderate

A

$Domain of f (x) is [- 1, 1]$
B

$Range of f (x) is [- \frac{2 π}{3}, \frac{π}{3}]$
C

$f (x) is one-one$
D

None of these

Solution

$Given 2 f (\sin x) + f (\cos x) = x - - - - - (1)$
$Replace x by \frac{π}{2} - x, we have 2 f (\cos x) + f (\sin x) = \frac{π}{2} - x - - - - - - (2)$
$\begin{aligned} Eliminating f (\cos x) from (1) and (2), we have f (\sin x) = x - \frac{π}{6} \\ \Rightarrow f (x) = \sin^{- 1} x - \frac{π}{6} \end{aligned}$
$Then, domain of f (x) is [- 1, 1] .$
$Range is [- \frac{π}{2} - \frac{π}{6}, \frac{π}{2} - \frac{π}{6}] or [- \frac{2 π}{3}, \frac{π}{3}]$
$Also, f (x) is one-one.$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App