# Thread: Find domain of function?

1. ## Find domain of function?

Find domain of the function:
$f(x) = \sqrt{\cos \left(\frac{\pi}{x^2}\right)}$

$\cos(\frac{\pi}{x^2})\geq 0$

$\frac{\pi}{x^2}\in[-\frac\pi2+2\pi n,\frac\pi2+2\pi n]$

$\frac1{x^2}\in[-\frac12+2n,\frac12+2n]$

$x^2\in[\frac1{\frac12+2n},\frac1{-\frac12+2n}]$

$x\in\pm [\frac1{\sqrt{\frac12+2n}},\frac1{\sqrt{-\frac12+2n}}]$

So domain $D=\cup_{n=1}^\infty [-\frac1{\sqrt{-\frac12+2n}},-\frac1{\sqrt{\frac12+2n}}]\cup[\frac1{\sqrt{\frac12+2n}},\frac1{\sqrt{-\frac12+2n}}]$