Question

Determine algebraically whether each function is even, odd, or neither. \(F(x)=\frac{2 x}{|x|}\)

Short answer

The function is odd.

Step-by-step solution

Understand the Definition of Even and Odd Functions

A function is even if for every x in its domain, the equality \(F(x) = F(-x)\) holds. A function is odd if for every x in its domain, the equality \(F(-x) = -F(x)\) holds.

Substitute -x into the Function

Given the function \(F(x) = \frac{2x}{|x|}\), substitute \(-x\) for \(x\) to find \(F(-x)\). The new expression is \(F(-x) = \frac{2(-x)}{| -x |}\).

Simplify the Expression for F(-x)

Simplify the expression \(F(-x) = \frac{2(-x)}{| -x |}\). Since \(| -x | = |x|\), the expression simplifies to \(F(-x) = \frac{-2x}{|x|}\).

Compare F(-x) with -F(x)

Calculate \(-F(x)\) for comparison: \(-F(x) = - \left( \frac{2x}{|x|} \right) = \frac{-2x}{|x|}\). Notice that \(F(-x) = \frac{-2x}{|x|}\) matches \(-F(x)\).

Determine if the Function is Odd

Since \(F(-x) = -F(x)\), the function \(F(x) = \frac{2x}{|x|}\) is classified as an odd function.

Key concepts

Function Classification

To understand functions better, it's important to classify them as even, odd, or neither.
This helps us understand their symmetrical properties and simplifies certain types of algebraic and calculus problems.
Here's how we classify functions:

• A function is **even** if for every X in its domain, the equality \(F(x) = F(-x)\) holds. This means the function is symmetrical about the Y-axis.
• A function is **odd** if for every X in its domain, the equality \(F(-x) = -F(x)\) holds. This means the function is symmetrical about the origin.
• A function is **neither** if it does not fit the criteria for being classified as even or odd.

By knowing these classifications, we can better understand the behavior of the function and predict its graphed shape more easily. This type of function categorization can often help in other branches of mathematics such as solving equations and analyzing graphs.

Absolute Value Functions

Absolute value functions are unique because they output the non-negative value of a number regardless of its sign.
Therefore, \(|x| = x \) if X is positive, and \(|x| = -x \) if X is negative.
This leads to interesting behavior in functions involving absolute values. For example, in the function \(F(x) = \frac{2x}{|x|}\), the absolute value function changes how the function behaves for both positive and negative values of X.

For positive x: \( F(x) = \frac{2x}{x} = 2 \)
For negative x: \( F(x) = \frac{2x}{-x} = -2 \)
The absolute value ensures that the denominator is always positive, while the numerator handles the sign. Understanding how absolute values work in functions helps make sense of these transformations and behaviors.

Algebraic Functions

Algebraic functions are constructed using algebraic operations with one or more variables. These functions can involve addition, subtraction, multiplication, division, and taking roots or powers.
They form the foundation of many mathematical concepts and include a wide variety of functions, from linear and quadratic functions to polynomials and rational functions.

In our exercise, the function \(F(x) = \frac{2x}{|x|}\) is a **rational function** combined with an absolute value.
Rational functions are any functions formed by the ratio of two polynomials.
This makes it essential to understand both the numerator and the denominator separately.
Simplifying and understanding these components helps in analyzing the function's behavior.

For example, since \( |x| = x \) or \( |x| = -x \), it influences the rational expression \(\frac{2x}{|x|}\) differently for positive and negative values.
Recognizing these patterns can help in determining whether the function is even, odd, or neither.