Trigonometry
Trigonometry
Allen Star Batch
Grade 11
Question:
If $f(x) = \sec^{-1}\left[1 + \cos^2 x\right]$, where $[.]$ denotes the greatest integer function, then :
The domain of $f$ is $\mathbb{R}$
The domain of $f$ is $[1, 2]$
The range of $f$ is $[1, 2]$
The range of $f$ is $\left\{\sec^{-1}1, \sec^{-1}2\right\}$
Step-by-Step Solution
Key Concept: The range of a reciprocal function is determined by the range of its denominator.
For $f(x) = \frac{1}{1+\cos^2 x}$ to be defined, we need $1+\cos^2 x \neq 0$, which is always true. Since $\cos^2 x \in [0,1]$, we have $1+\cos^2 x \in [1,2]$, so $f(x) = \frac{1}{1+\cos^2 x} \in [\frac{1}{2}, 1]$. Therefore, the domain of $f$ is $\mathbb{R}$ and the range is $[\sec^{-1}1, \sec^{-1}2]$.
Correct Answer: 1,4