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The Square Root Function

The square root function y = the square root of x starts at the origin and curves upward, defined only for x greater than or equal to 0 since negative numbers have no real square root. Learn its domain and range (both x,y greater than or equal to 0), key points, and how it relates to a sideways parabola.

The square root function

The square root function, y = √x, takes each non-negative input and returns its principal (positive) square root. Its graph starts at the origin and curves gently upward and to the right, rising more slowly as x grows.

y = √x starts at the origin and is only defined for x ≥ 0.

Domain and range

Since a negative number has no real square root, the domain is restricted to x ≥ 0. The output of a square root is never negative, so the range is also y ≥ 0. Both domain and range start at 0 and extend to positive infinity.

Key points

A few values anchor the shape: √0 = 0, √1 = 1, √4 = 2, √9 = 3. Notice that as x quadruples, y only doubles — that's why the curve flattens out as it moves right, instead of continuing to rise steeply like a line or parabola.

Related shapes

The square root graph is half of a sideways parabola: squaring both sides of y = √x gives y² = x, the equation of a horizontal parabola, but restricting y to non-negative values keeps only the upper half. Shifting, stretching, or reflecting this base shape leads to transformations of radical functions, and solving equations that contain a square root is covered in solving radical equations.

Related lessons

Algebra