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Find the domain of the function \(f(x) = \dfrac{1}{2 \sqrt{\sqrt{x} - x} - 1}\)

 Jun 9, 2020
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The domain of the function consists of all real numbers except those numbers that make the denominator equal to zero.

Let's try to find those numbers.

 

2·sqrt[  sqrt(x) - x  ] - 1  =  0

     2·sqrt[  sqrt(x) - x  ]  =  1

        sqrt[  sqrt(x) - x  ]  =  ½

                    sqrt(x) - x  =  ¼

            -x + sqrt(x) - ¼  =  0

             x - sqrt(x) + ¼  =  0

 

Using the quadratic equation:  sqrt(x)  =  [ -(-1) +/- sqrt( (1)2 - 4·1·¼ ) ] / (2·1)

                                                 sqrt(x)  =  [ 1 + sqrt(0) ] / 2

                                                 sqrt(x)  =  ½

                                                          x  =  ¼

 

¼ is the number that cannot be included in the domain; so the domain consists of all real numbers except ¼.

 Jun 10, 2020

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