key term - Even Root Function

Definition

An even root function is a type of radical function where the index of the radical is an even number, such as 2, 4, or 6. These functions have the general form $f(x) = \sqrt[n]{x}$, where $n$ is an even integer. Even root functions are characterized by their symmetric nature about the $y$-axis and their ability to accept both positive and negative input values.

5 Must Know Facts For Your Next Test

1. Even root functions are symmetric about the $y$-axis, meaning that $f(-x) = f(x)$.
2. The domain of an even root function is all real numbers greater than or equal to 0, $x \geq 0$.
3. The range of an even root function is all real numbers greater than or equal to 0, $y \geq 0$.
4. Even root functions are invertible, and their inverse functions are also even root functions.
5. The graphs of even root functions are smooth, continuous, and increasing on their domain.

Review Questions

• Explain the symmetry property of even root functions and how it relates to their domain and range.
  • Even root functions, such as $f(x) = \sqrt{x}$ or $f(x) = \sqrt[4]{x}$, are symmetric about the $y$-axis. This means that the function values for positive and negative input values are the same, i.e., $f(-x) = f(x)$. This symmetry is a direct consequence of the even exponent in the radical expression. The domain of an even root function is all non-negative real numbers, $x \geq 0$, since negative inputs would result in complex outputs, which are not part of the real number system. Similarly, the range of an even root function is also all non-negative real numbers, $y \geq 0$, due to the nature of the radical function.
• Describe the relationship between even root functions and their inverse functions.
  • Even root functions are invertible, meaning that for every output value, there is a unique input value that produces that output. The inverse function of an even root function is also an even root function, but with the roles of the input and output variables reversed. For example, if $f(x) = \sqrt{x}$, then the inverse function is $f^{-1}(x) = x^2$, which is also an even root function. This relationship between even root functions and their inverses is a key property that is often exploited when working with radical functions and their applications.
• Analyze the graph of an even root function and explain how its properties, such as domain, range, and symmetry, are reflected in the visual representation.
  • The graph of an even root function is smooth, continuous, and increasing on its domain, which is all non-negative real numbers. The graph is symmetric about the $y$-axis, meaning that the function values for positive and negative input values are the same. This symmetry is a direct consequence of the even exponent in the radical expression. The domain of the function is reflected in the $x$-axis, where the graph is defined for all $x \geq 0$. Similarly, the range of the function is reflected in the $y$-axis, where the graph is defined for all $y \geq 0$. The smooth, continuous, and increasing nature of the graph is a result of the properties of the radical function, which is a monotonically increasing function on its domain.

Related terms

Radical Function:

A radical function is a function that contains a radical expression, such as a square root or higher-order root.

Inverse Function:

An inverse function is a function that undoes the operation of another function, reversing the input and output values.

Domain and Range:

The domain of a function is the set of input values, and the range is the set of output values that the function can produce.