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Thread: Find domain of function?

  1. #1
    Super Member fardeen_gen's Avatar
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    Find domain of function?

    Find domain of the function:
    $\displaystyle f(x) = \sqrt{\cos \left(\frac{\pi}{x^2}\right)}$
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  2. #2
    Senior Member
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    Atlanta, GA
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    Not so bad...

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

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

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

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

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

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