First slide

Introduction to ITF

Question

$Range of f (x) = \sin^{- 1} x + \tan^{- 1} x + \sec^{- 1} x is$

Moderate

A

$(\frac{π}{4}, \frac{3 π}{4})$
B

$[\frac{π}{4}, \frac{3 π}{4}]$
C

$\{\frac{π}{4}, \frac{3 π}{4}\}$
D

None of these

Solution

$We have, f (x) = \sin^{- 1} x + \tan^{- 1} x + \sec^{- 1} x Domain of \sin^{- 1} x is [- 1, 1] Domain of \tan^{- 1} x is (- \infty, \infty) Domain of \sec^{- 1} x is (- \infty, - 1] \cup [1, \infty) ∴ Domain of f (x) = {- 1, 1} Thus, range of f (x) = {f (1), f (- 1)} i . e . \sin^{- 1} 1 + \tan^{- 1} 1 + \sec^{- 1} 1 = \frac{π}{2} + \frac{π}{4} + 0 = \frac{3 π}{4} and \sin^{- 1} (- 1) + \tan^{- 1} (- 1) + \sec^{- 1} (- 1) = - \frac{π}{2} - \frac{π}{4} + π = \frac{π}{4} ∴ Range is \{\frac{π}{4}, \frac{3 π}{4}\} .$

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