Question

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

Answer 1

`x in ...U [ -sqrt((5pi)/2), -sqrt((3pi)/2) ] U [ -sqrt(pi/2), sqrt(pi/2) ] U [ sqrt((3pi)/2), sqrt((5pi)/2) ] U [ sqrt((7pi)/2), sqrt((9pi)/2) ] U ...` and `y in [ 0, 1 ]`

Explanation:

`0 <= cos ( x^2 ) <= 1`, and so,

`0 <= y = sqrt cos (x^2 ) <= 1`. Inversely,

`x = +- sqrt( abs( cos^(-1)y^2)), +- sqrt(2kpi +- cos^(-1)y^2)`,

`k = 1, 2, 3, ...`, avoiding negatives under the radical sign.

So,

`x in ...U [ -sqrt((9pi)/2), -sqrt((7pi)/2) ] U`

`[ -sqrt((5pi)/2), sqrt((3pi)/2) ] U [ -sqrt(pi/2), sqrt(pi/2) ]`

`U [ sqrt((3pi)/2), sqrt((5pi)/2) ] U [sqrt((7pi)/2), sqrt((9pi)/2) ]U...`

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