Question

The Range of the function $y={(\frac{co{s}^{-1}(3x-1)}{\pi }+1)}^2$ is

• A. [1, 4]
• B. [0, $\pi$]
• C. [1, $\pi$]
• D. [0, ${\pi }^2$]

Answer 1

The correct option is A [1, 4]

We know that $co{s}^{-1}\left(x\right)$ has a range of $[0,\pi ]$
Thus
$0\le co{s}^{-1}(3x-1)\le \pi$
$0\le \frac{co{s}^{-1}(3x-1)}{\pi }\le 1$
$0+1\le \frac{co{s}^{-1}(3x-1)}{\pi }+1\le 1+1$
$1\le \frac{co{s}^{-1}(3x-1)}{\pi }+1\le 2$
$1^2\le (\frac{co{s}^{-1}(3x-1)}{\pi }+1{)}^2\le 4$
$1\le y\le 4$
Hence the range is $[1,4]$