sqrtX2

sqrtx2 commonly denotes the expression sqrt(x^2). For real x, this value equals the absolute value of x, written |x|. This is because squaring x makes its sign disappear, and taking the principal (nonnegative) square root returns the magnitude. Consequently, sqrt(x^2) = x when x is nonnegative, and sqrt(x^2) = -x when x is negative.

As a function, y = sqrt(x^2) is the absolute value function, y = |x|. Its graph is a V shape with vertex at the origin, symmetric about the y-axis. The domain is all real numbers, and the range is [0, ∞). The expression is never negative.

Key distinctions are important. sqrt(x^2) is not generally equal to x; they agree only for x ≥ 0. By contrast, sqrt(x)^2 equals x for all x ≥ 0 in the real numbers, but sqrt(x) itself is not defined for x < 0 in reals. Thus sqrt(x^2) and (sqrt(x))^2 are different operations with different domains and interpretations.

In complex numbers, the square root is multivalued, so sqrt(z^2) need not equal z or |z}. The principal branch of the square root introduces a choice, and simple identities from the real case may fail in the complex plane.

Examples: sqrt(9) = 3; sqrt(4) = 2; sqrt((-3)^2) = sqrt(9) = 3. The result remains nonnegative, reflecting the definition of the principal square root.