Radical Functions — Mathematics Knowledge System

Overview

A radical function contains its variable inside a root, most commonly a square root: f(x) = √(x). Radical functions arise as the inverses of power functions (√x undoes squaring on x≥0), and their domain and range are restricted by the requirement that expressions under an even root be nonnegative.

Interactive Graph

Formal Definition

Definition

The basic square root and n-th root functions:

f(x) = \sqrt[n]{g(x)}

General radical function

\text{dom}(f) = \{x : g(x) \geq 0\} \text{ when } n \text{ is even}

Even roots require a nonnegative radicand; odd roots (cube roots, etc.) allow all reals

Properties

Domain of √x

\text{dom} = [0,\infty)

Range of √x

\text{ran} = [0,\infty)

Domain of ∛x

\text{dom} = \mathbb{R} \text{ (odd roots accept negative inputs)}

Inverse relationship

f(x) = \sqrt{x} \text{ is the inverse of } g(x) = x^2 \text{ restricted to } x \geq 0

Worked Examples

The radicand must be nonnegative since this is a square (even) root.
$3x-6 \geq 0 \implies x \geq 2$

Answer: Domain: x ≥ 2, i.e. [2, ∞)

Practice Problems

Difficulty 3/10

Find the domain and range of f(x) = √(x+4) - 1.

Common Mistakes

Common Mistake

Assuming all radical functions have domain restrictions.

Only EVEN-index roots (square roots, 4th roots, ...) require a nonnegative radicand. Odd-index roots like cube roots accept negative numbers just fine (∛(-8) = -2), so f(x)=∛x has domain all reals.

Summary

A radical function has its variable under a root: f(x) = ⁿ√g(x).
For even n, the domain requires g(x) ≥ 0; for odd n, the domain is all reals.
f(x)=√x is the inverse of x² restricted to x≥0, with domain and range both [0,∞).

References

WebsiteWikipedia — Nth root