Find the domain of the function \(f(x) = \dfrac{1}{2 \sqrt{\sqrt{x} - x} - 1}\)
+23246 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.
