Radical functions

Definition

A radical function is a function that involves a root, typically the square root, cube root, or higher-order roots of a variable. These functions are expressed in the form $f(x) = \sqrt[n]{x}$ where $n$ is the index of the root.

5 Must Know Facts For Your Next Test

1. The domain of a radical function depends on the index of the root; for even roots like square roots, the radicand must be non-negative.
2. Radical functions can be inverted if they are one-to-one; their inverses will often involve raising to a power instead of taking a root.
3. Simplifying and rationalizing expressions involving radicals are common operations when working with these functions.
4. Graphing radical functions involves understanding their transformations such as shifts, reflections, and stretches/compressions.
5. $\text{For example, } f(x) = \sqrt{x-2} +3 \text{ represents a right shift by 2 units and an upward shift by 3 units.}$

Review Questions

• What is the domain of the function $f(x) = \sqrt{x+4}$?
• How would you graphically represent $g(x) = \sqrt[3]{x-1} - 2$?
• Explain how to find the inverse of $h(x) = \sqrt{x - 7}$.

Related terms

Square Root Function: A specific type of radical function where the index of the root is two, typically written as $f(x) = \sqrt{x}$.

Inverse Function: A function that reverses another function; if $f(x)$ maps $x$ to $y$, then its inverse $f^{-1}(y)$ maps $y$ back to $x$.

Rational Exponent: $An exponent written as a fraction; for instance, x^{1/n} is equivalent to \sqrt[n]{x}.$