First slide

Functions (XII)

Question

Let $f (x) = \sec^{- 1} [1 + \cos^{2} x]$ , where $[.]$ denotes the greatest integer function. Then the

Moderate

A

domain of f is R
B

domain of f is [1, 2]
C

domain of f is [2, 3]
D

range of f is $\{\sec^{- 1} 1, \sec^{- 1} 2\}$

Solution

$\begin{array}{l} f (x) = \sec^{- 1} [1 + \cos^{2} x] \\ f (x) is defined if [1 + \cos^{2} x] \leq - 1 or [1 + \cos^{2} x] \geq 1 \end{array}$
$\begin{array}{l} i.e., [\cos^{2} x] \leq - 2 (not possible) or [\cos^{2} x] \geq 0 \\ i.e., \cos^{2} x \geq 0 or x \in R \\ Now, 0 \leq \cos^{2} x \leq 1 or 1 \leq 1 + \cos^{2} x \leq 2 \\ or [1 + \cos^{2} x] = 1, 2 \\ or \sec^{- 1} [1 + \cos^{2} x] = \sec^{- 1} 1, \sec^{- 1} 2 \\ Hence, the range is \{\sec^{- 1} 1, \sec^{- 1} 2\} \end{array}$

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