Function library
Square Root Function
f(x) = sqrt(x)
The gentle half-parabola — the inverse of squaring, defined for x ≥ 0.
The square root function f(x) = √x answers “which non-negative number, multiplied by itself, gives x?” Its graph is the upper half of a sideways parabola: starting at the origin, it rises steeply — with a vertical tangent — then bends over and flattens, passing through (1, 1), (4, 2), and (9, 3). It is the inverse of x² restricted to x ≥ 0, so its graph is the mirror of the right half of the parabola y = x² reflected across the line y = x.
Square roots appear wherever Pythagoras does: the distance formula √((Δx)² + (Δy)²) is a square root, and so are standard deviation, the quadratic formula’s discriminant term, and the root-mean-square behind AC voltage ratings. The function grows without bound but ever more slowly — √1,000,000 is only 1,000. Type sqrt(x) into the calculator and plot x^2 on [0, ∞) beside it to see the mirror symmetry.
Computed properties
Calculated by the graphing calculator's own math engine at build time for sqrt(x).
- Roots in [−10, 10]
- 0.0000
- y-intercept f(0)
- 0.0000
- Local extrema in [−10, 10]
- none found
- Sample values
- f(-2) = undefined · f(-1) = undefined · f(0) = 0 · f(1) = 1 · f(2) = 1.4142
- Derivative f′(1)
- 0.5000
- Integral ∫₀¹ f(x) dx
- undefined
Key facts
- Principal root: √x ≥ 0 for all x in the domain; √9 = 3, not ±3.
- Domain: x ≥ 0; range: y ≥ 0. Undefined for negative x (over the reals).
- Inverse of x² on [0, ∞): √(x²) = |x|, and (√x)² = x for x ≥ 0.
- Key points: (0, 0), (1, 1), (4, 2), (9, 3); vertical tangent at the origin.
- Increasing and concave down on its domain; minimum 0 at x = 0, no maximum.
- Derivative d/dx √x = 1/(2√x); laws √(ab) = √a·√b for a, b ≥ 0.
What the square root is
The symbol √ denotes the principal (non-negative) square root: √9 = 3, not ±3, because a function must give one output per input. The equation x² = 9 has two solutions, ±3, but the function √x returns only the non-negative one — the ± belongs to solving equations, not to the function. This single-valuedness is what makes √x differentiable and graphable as one clean curve.
Algebraically, √(x²) = |x|, not x — the root undoes the square but restores non-negativity, which is why the absolute-value function appears. The root also obeys √(ab) = √a·√b and √(a/b) = √a/√b for non-negative a, b, the multiplicative laws inherited from exponents since √x = x^(1/2).
Domain, shape, and the vertical tangent
The domain is x ≥ 0: no real number squares to a negative, so the function cannot accept negative inputs (over the reals). At x = 0 the graph starts with a vertical tangent — the derivative 1/(2√x) blows up to +∞ — which you can see as the curve leaving the origin straight upward before bending right. It is increasing everywhere on its domain and concave down everywhere, flattening as x grows.
The range is y ≥ 0, and notable points are perfect squares: (0, 0), (1, 1), (4, 2), (9, 3), (16, 4). Between them the curve interpolates smoothly — √2 ≈ 1.414, the famous irrational whose discovery shook Pythagorean mathematics. The function has no maximum and its only endpoint extremum is the minimum 0 at x = 0.
Where square roots appear
Distance is the square root’s home turf: from Pythagoras’ hypotenuse to the n-dimensional distance formula to the standard deviation (the square root of variance), “square, add, root” is one of mathematics’ most repeated patterns. The quadratic formula x = (−b ± √(b² − 4ac))/(2a) puts a square root at the heart of solving quadratics — including finding where x² − 4 crosses zero.
In physics, many laws involve square roots: the period of a pendulum is proportional to √(length), escape velocity to √(1/radius), and the RMS voltage of AC mains is the peak voltage divided by √2. In geometry, √2 is the diagonal of a unit square and the aspect ratio of A-series paper.
Frequently asked questions
Why isn’t √9 equal to ±3?
Because √ denotes a function, and functions return exactly one value per input — by convention the non-negative root. The equation x² = 9 does have two solutions, x = 3 and x = −3, but only 3 is √9. Writing ±√9 recovers both solutions when solving.
Why can’t you take the square root of a negative number (in reals)?
Because every real number squared is non-negative: positives give positive squares, negatives give positive squares, and zero gives zero. Nothing real squares to −1, so √(−1) has no real value. Extending the number system with i, where i² = −1, gives complex square roots.
What is the derivative of √x at x = 0?
It doesn’t exist — the derivative 1/(2√x) tends to +∞ as x → 0⁺, so the graph has a vertical tangent at the origin. Geometrically the curve leaves (0, 0) heading straight up; there is no finite slope there, though the function itself is continuous at 0.