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How would you find the domain for f(x) = 2x - 4x^1/2 + 2 ?

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The domain of this function is all real numbers greater than or equal to zero.  The range is 2 to + infinity.
Negative real numbers are excluded from the domain because the square root of a negative number is not real.   The minimum value of the function occurs at x = 0  and has the value 2.   The 2 x term ensures that the upper end of the range is + infinity.