Root and Radical Functions
Root functions, also called radical functions, are functions in which the variable appears under a root sign, such as f(x) = √x or f(x) = ∛x. In brief: the graph of y = √x starts at the origin and rises slowly to the right, with domain x ≥ 0 and range y ≥ 0. The graph of y = ∛x passes through the origin, is defined for all real x and is symmetric about the origin.
Root functions are the inverses of power functions, and they describe slow, decelerating growth, such as the time an object takes to fall a given distance. This guide shows the graphs of the basic root functions, how to find domain and range, how to apply transformations and how to solve typical problems.
What does the graph of y = √x look like?
The square root function f(x) = √x is defined only for x ≥ 0, since negative numbers have no real square root. Its graph starts at the point (0, 0) and rises to the right, but the slope keeps decreasing: the curve is steep near 0 and flattens as x grows. It is half of a parabola lying on its side.
Key points are (0, 0), (1, 1), (4, 2) and (9, 3). The function is increasing on its whole domain, has no maximum and its values grow without bound, though slowly. It is not symmetric: its left end is closed at the origin.
What does the graph of y = ∛x look like?
The cube root function f(x) = ∛x is defined for all real numbers. It passes through (−8, −2), (−1, −1), (0, 0), (1, 1) and (8, 2). The graph is steepest at the origin, where the tangent is vertical, and flattens as |x| increases.
The function is odd: ∛(−x) = −∛x, so the graph is symmetric with respect to the origin. It is increasing everywhere and its range is all real numbers. Higher roots behave like these two: even roots resemble √x, and odd roots resemble ∛x.
Value table for root functions
| x | √x | ∛x |
|---|---|---|
| −27 | not real | −3 |
| −8 | not real | −2 |
| −1 | not real | −1 |
| 0 | 0 | 0 |
| 1 | 1 | 1 |
| 4 | 2 | 1.587 (approx.) |
| 8 | 2.828 (approx.) | 2 |
| 9 | 3 | 2.080 (approx.) |
| 27 | 5.196 (approx.) | 3 |
How do you find the domain and range?
- For an even root, set the radicand ≥ 0 and solve the inequality. This is the domain.
- For an odd root, the domain is all real numbers unless a denominator or other restriction applies.
- Find the range from the lowest or highest value of the root expression, then apply any outside shift or stretch.
Example: f(x) = √(2x − 6) + 1
Require 2x − 6 ≥ 0, so x ≥ 3. The domain is [3, ∞). The root is at least 0, so f(x) ≥ 1, and the range is [1, ∞). The starting point is (3, 1).
How do transformations change a root graph?
The general form is y = a√(x − h) + k. The starting point is (h, k). The number h shifts the graph horizontally, k shifts it vertically and a stretches it vertically. If a is negative, the graph reflects across the horizontal line through the start point. A negative sign inside, y = √(−x), reflects the graph across the y-axis and gives domain x ≤ 0.
Example: y = −√(x + 2) + 3
The start point is (−2, 3). The graph goes downward to the right because of the negative sign. The domain is x ≥ −2 and the range is y ≤ 3. At x = 2, y = −√4 + 3 = 1.
How are root functions related to power functions?
The function y = √x is the inverse of y = x2 restricted to x ≥ 0. Reflecting the right branch of the parabola in the line y = x gives the graph of the square root. Similarly, y = ∛x is the inverse of y = x3. The restriction is needed because the full parabola fails the one-to-one test, so only half of it has an inverse.
Where do root functions come from?
The shape of y = √x is that of a parabola turned on its side, a curve that Greek mathematicians such as Apollonius studied as a conic section. Considering it as a function came much later, after the notion of a function was developed in the seventeenth and eighteenth centuries. The relation to inverse functions was formalised in the same period.
Where are root functions used today?
The time t for an object to fall from rest through a height h is t = √(2h/g), which is a square root function of h. For h = 20 m and g ≈ 9.8 m/s2, t ≈ 2.0 s. The period of a pendulum of length L is proportional to √L, so quadrupling the length doubles the period. The speed of a wave on a string depends on a square root of the tension. The side length of a square from its area is also √A.
Which mistakes are common?
Ignoring the domain
√(x − 4) is undefined for x < 4. Always solve the radicand inequality first.
Shifting the wrong way
√(x + 2) starts at x = −2, which is a shift to the left, not the right.
Drawing a smooth start
The graph of y = √x has a vertical tangent at the origin. Do not draw it with a flat start.
Confusing √x with x2
√x grows slowly and x2 grows fast. For x = 16, √x = 4 and x2 = 256.