Radical Function — Definition, Formula & Examples

A radical function is a function whose rule has the input variable inside a root, such as a square root or a cube root. The graph of a square root function starts at a single endpoint and curves away in one direction. A cube root function forms a smooth S-shape.

A radical function is a function of the form $f(x) = a\sqrt[n]{g(x)} + k$ , where $n \ge 2$ is an integer called the index, $a \ne 0$ , and the radicand $g(x)$ contains the variable. When $n$ is even, the domain is the set of all $x$ with $g(x) \ge 0$ . When $n$ is odd, the domain is every $x$ for which $g(x)$ is defined.

Key Formula

f(x) = a\sqrt[n]{x - h} + k

Where:

$a$ = Vertical stretch factor. A negative value reflects the graph across the horizontal line y = k.
$n$ = Index of the root, an integer with n ≥ 2 (n = 2 gives a square root)
$h$ = Horizontal shift. The graph moves right by h units.
$k$ = Vertical shift. The graph moves up by k units.
$(h, k)$ = Endpoint of the graph when n is even. It is the point of symmetry when n is odd.

Worked Example

Problem: Find the domain and range of

f(x) = \sqrt{x - 4} + 2

. Then evaluate

f(13)

Step 1: The index is 2, which is even. So the radicand must be nonnegative. Solve the inequality for x.

x - 4 \ge 0 \quad\Longrightarrow\quad x \ge 4

Step 2: Evaluate f at the endpoint x = 4. A square root is never negative, so the smallest output occurs here. Every other output is greater.

f(4) = \sqrt{0} + 2 = 2 \quad\Longrightarrow\quad y \ge 2

Step 3: Substitute x = 13 into the function.

f(13) = \sqrt{13 - 4} + 2 = \sqrt{9} + 2 = 3 + 2 = 5

Answer: The domain is

x \ge 4

, or

[4, \infty)

. The range is

y \ge 2

, or

[2, \infty)

. The value of the function at 13 is

f(13) = 5

Why It Matters

In physics, a pendulum's period is the radical function

T = 2\pi\sqrt{L/g}

of its length. That is why quadrupling the length only doubles the period. In Algebra 2 and precalculus, square root functions also appear as inverses of quadratic functions whose domain has been restricted.

Common Mistakes

Mistake: Assuming every radical function has a restricted domain.

Correction: Only even-index roots require a nonnegative radicand. A cube root accepts negative inputs, because

\sqrt[3]{-8} = -2

. So

f(x) = \sqrt[3]{x - 4}

has a domain of all real numbers.

Related Terms

Function — the broader concept this term builds on
Algebraic Function — the broader concept this term builds on
Absolute Value Function — Another parent function transformed by a, h, k
Continuous Function — Radical functions are continuous on their domains
Bounded Function — Square root functions are bounded on one side
Decreasing Function — A negative a makes the graph decrease
Dependent Variable — The output f(x) produced from each input