Question

Let $f\left(x\right)=se{c}^{-1}[1+co{s}^{2}x]$, where [.] denotes the greatest integer function, then the range of f(x) is

• A. [1, 2]
• B. [0, 2]
• C. $[se{c}^{-1}1,se{c}^{-1}2]$
• D. None of these

Answer 1

The correct option is C $[se{c}^{-1}1,se{c}^{-1}2]$

$f\left(x\right)=se{c}^{-1}\{1+co{s}^{2}x\}0\le co{s}^{2}x\le 10+1\le 1+co{s}^{2}x\le 1+11\le 1+co{s}^{2}x\le 2$
$[1+co{s}^{2}x]=1$ ([] is the greatest integer function)
$(for1\le (1+co{s}^{2}x)<2)$
$[1+co{s}^{2}x]=2$
$\left(for\right(1+co{s}^{2}x)=2)$
$\therefore$ the values of
$y=se{c}^{-1}[1+co{s}^{2}x]arese{c}^{-1}1$ and $se{c}^{-1}2$