Question

Determine algebraically whether each function is even, odd, or neither. \(G(x)=\sqrt{x}\)

Short answer

Neither even nor odd.

Step-by-step solution

- Understand the Definitions

First, let's recall the definitions of even, odd, and neither functions. A function is even if it satisfies the condition: \( f(-x) = f(x) \) for all x in the domain. A function is odd if it satisfies the condition: \( f(-x) = -f(x) \) for all x in the domain. If a function does not satisfy either of these conditions, it's classified as neither.

- Substitute -x into the Function

Next, substitute \(-x\) into the function \(G(x)\). So, calculate \(G(-x)\). For the given function \(G(x) = \sqrt{x}\), we get: \(G(-x) = \sqrt{-x}\).

- Compare with the Original Function

Now, compare \(G(-x)\) with \(G(x)\). Check if \(G(-x) = G(x)\) or \(G(-x) = -G(x)\). In this case, \(G(x) = \sqrt{x}\) and \(G(-x) = \sqrt{-x}\), which does not simplify to \(G(x)\) or \(-G(x)\). Notice that \(\sqrt{-x}\) is not defined for all real numbers x.

- Conclude the Nature of the Function

Since \(G(-x)\) does not equal \(G(x)\) or \(-G(x)\), \(G(x)\) is neither even nor odd.

Key concepts

function analysis

Function analysis involves breaking down a function step by step to understand its properties and behavior. For our exercise, we're examining the function \( G(x) = \sqrt{x} \) to determine whether it is even, odd, or neither.

To start, we substitute \( -x \) into the function and then compare the result with the original function.

Substituting \( -x \) in \( G(x) \) gives us \( G(-x) = \sqrt{-x} \).

Next, we compare \( G(-x) \) with \( G(x) \). In this case, \( G(-x) \) does not simplify to either \( G(x) \) or \( -G(x) \). Additionally, \( \sqrt{-x} \) isn't defined for all real numbers \( x \).

This analysis leads us to conclude that \( G(x) = \sqrt{x} \) is neither even nor odd.

function properties

Understanding the properties of a function helps determine how it behaves across its domain. Key properties include whether a function is even, odd, or neither.

An even function satisfies \( f(-x) = f(x) \), meaning it is symmetric about the y-axis. Examples include \( f(x) = x^2 \) and \( f(x) = \cos(x) \).

An odd function satisfies \( f(-x) = -f(x) \) and is symmetric about the origin. Examples include \( f(x) = x^3 \) and \( f(x) = \sin(x) \).

If a function does not satisfy either condition, it is classified as neither even nor odd. Analyzing these properties provides deeper insight into function behavior and aids in graphing.

algebraic functions

Algebraic functions are built from algebraic operations like addition, subtraction, multiplication, division, and taking roots. They are expressed using polynomials, rational functions, and root functions.

The function \( G(x) = \sqrt{x} \) is a root function, specifically a square root function.

Algebraic functions can have varied properties and behaviors. For example, \( f(x) = x^2 \) is even, while \( f(x) = x^3 \) is odd.

Recognizing the type of algebraic function helps in analyzing and determining its properties, including whether it is even or odd.

domain of a function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. It is essential to consider the domain when analyzing a function's properties.

For \( G(x) = \sqrt{x} \), the domain includes all non-negative real numbers (\( x \geq 0 \)), as the square root function is not defined for negative values.

When determining whether a function is even or odd, it's crucial to ensure that the domain is symmetric about the origin. If this is not the case, the function cannot be classified as even or odd.

Since \( G(x) \) is not defined for negative \( x \) values, it cannot exhibit even or odd properties over its domain, supporting the conclusion that \( G(x) \) is neither even nor odd.