Question

Let $f\left(x\right)=se{c}^{-1}[1+co{s}^{2}x]$ where $[\cdot ]$ denotes the greatest integer function Then

• A. the domain of $f$ is $R$
• B. the domain of $f$ is $[1,2]$
• C. the range$f$ is $[1,2]$
• D. the range of $f$ is $\{se{c}^{-1}1,se{c}^{-1}2\}$

Answer 1

Answer: (A), (D)

the domain of $f$ is $R$
the range of $f$ is $\{se{c}^{-1}1,se{c}^{-1}2\}$

Note that $-1\le \cos \left(x\right)\le 1$, we have $0\le {\cos }^2\left(x\right)\le 1$.
Hence the only possible values for $\left[1+{\cos }^2\left(x\right)\right]$ are 1 and 2.
As $\left[1+{\cos }^2\left(x\right)\right]=1$ if ${\cos }^2\left(x\right)<1$ and
$\left[1+{\cos }^2\left(x\right)\right]=2$ if ${\cos }^2\left(x\right)=1$.
Hence the range of the function is $\left\{{\sec }^{-1}1,{\sec }^{-2}2\right\}$.
Since, $\cos \left(x\right)$ is defined for all real numbers, so is the given function. Hence the domain of the function $f\left(x\right)$ is $R$.