What is the domain and range of sqrtcos(x^2)?

Jul 4, 2018

$x \in \ldots U \left[- \sqrt{\frac{5 \pi}{2}} , - \sqrt{\frac{3 \pi}{2}}\right] U \left[- \sqrt{\frac{\pi}{2}} , \sqrt{\frac{\pi}{2}}\right] U \left[\sqrt{\frac{3 \pi}{2}} , \sqrt{\frac{5 \pi}{2}}\right] U \left[\sqrt{\frac{7 \pi}{2}} , \sqrt{\frac{9 \pi}{2}}\right] U \ldots$ and $y \in \left[0 , 1\right]$

Explanation:

$0 \le \cos \left({x}^{2}\right) \le 1$, and so,

$0 \le y = \sqrt{\cos} \left({x}^{2}\right) \le 1$. Inversely,

$x = \pm \sqrt{\left\mid {\cos}^{- 1} {y}^{2} \right\mid} , \pm \sqrt{2 k \pi \pm {\cos}^{- 1} {y}^{2}}$,

$k = 1 , 2 , 3 , \ldots$, avoiding negatives under the radical sign.

So,

$x \in \ldots U \left[- \sqrt{\frac{9 \pi}{2}} , - \sqrt{\frac{7 \pi}{2}}\right] U$

$\left[- \sqrt{\frac{5 \pi}{2}} , \sqrt{\frac{3 \pi}{2}}\right] U \left[- \sqrt{\frac{\pi}{2}} , \sqrt{\frac{\pi}{2}}\right]$

$U \left[\sqrt{\frac{3 \pi}{2}} , \sqrt{\frac{5 \pi}{2}}\right] U \left[\sqrt{\frac{7 \pi}{2}} , \sqrt{\frac{9 \pi}{2}}\right] U \ldots$

See the phenomenal graph.
graph{y - sqrt(cos(x^2))=0[-10 10 -1 9]}