sqrt2x

sqrt2x commonly denotes the square root of the product 2x, written as sqrt(2x). For real numbers, this expression is defined when x is nonnegative (x ≥ 0). In that domain, sqrt(2x) is the nonnegative square root of 2x and satisfies sqrt(2x) = (2x)^{1/2}. It can also be written as sqrt(2) sqrt(x) for x ≥ 0.

Domain and range: the real-valued function f(x) = sqrt(2x) has domain [0, ∞) and range [0, ∞). It is continuous and increasing on its domain. The graph is a concave-down curve on x > 0.

Calculus: the derivative is f'(x) = 1 / sqrt(2x) for x > 0, with a vertical tangent at x = 0. The indefinite integral is ∫ sqrt(2x) dx = (1/3) (2x)^{3/2} + C.

Algebraic properties: for x ≥ 0, sqrt(2x) can be factored as sqrt(2) sqrt(x). Squaring both sides gives (sqrt(2x))^2 = 2x, which holds in the real domain.

Examples: x = 0 gives sqrt(0) = 0; x = 1/2 gives sqrt(1) = 1; x = 2 gives sqrt(4) = 2.

Notes: In the complex plane, sqrt(2x) can be extended via branches, with sqrt(2x) = sqrt(2) sqrt(x) on appropriate branches and considering possible sign choices. Real-valued sqrt(2x) remains defined only for x ≥ 0.