Problem 39 - Precalculus : Building Concepts and Connections

Question

Decide if each function is odd, even, or neither by using the definitions. $$f(x)=2 x$$

Step-by-Step Solution

Answer

The function \(f(x)=2x\) is odd.

Step 1: Check if the function is even

Substitute \(x\) with \(-x\) in function \(f(x)=2x\). This results in \(f(-x)=2(-x)=-2x\). The result is not equal to the original function \(f(x)=2x\), so the function is not even.

Step 2: Check if the function is odd

Check if the negation of the function, \(-f(x)\), equals \(f(-x)\). The negation of \(f(x)\) is \(-f(x)=-2x\). Since \(-f(x)=f(-x)\), we can conclude that the function is odd.

Key Concepts

Function Symmetry

Understanding function symmetry helps us classify functions as either even, odd, or neither. Symmetry in mathematics essentially means that something looks the same when viewed from different perspectives or transformations. In the context of functions, symmetry refers to how the graph of a function behaves when it is reflected over specific lines.

Here's a quick guide to function symmetry:

• **Even functions** have symmetry across the y-axis. Think of a mirror placed on the y-axis. If both sides of the graph are identical reflections, the function is even.


• **Odd functions** have rotational symmetry around the origin, meaning you can rotate the graph by 180 degrees around the origin, and it will look the same.


• **Neither** functions won’t display either type of symmetry when graphed.

Identifying symmetry can be visually realized through graphing or algebraically by modifying function expressions.

Even Function Definition

The definition of even functions centers around their symmetry with respect to the y-axis. This means that flipping the input variable won't change the value of the function. Mathematically, a function is even if:

• For every integer value of x:
  \[ f(-x) = f(x) \]

To illustrate, if you have a function like \( f(x) = x^2 \), substituting \( x \) with \( -x \) gives you:

• \( f(-x) = (-x)^2 = x^2 \)

Since \( f(-x) = f(x) \), this confirms that \( x^2 \) is an even function.Even functions often feature powers of x raised to an even number (e.g., \(x^2, x^4\)). However, not all polynomial functions with even powers are strictly even due to potential mixed terms.

Odd Function Definition

The definition of odd functions revolves around rotational symmetry, specifically at the origin. In simple terms, it implies that if you take every positive input and reflect it negatively across the origin, the function's value changes sign, while maintaining the same magnitude. To be mathematically precise:

• A function is odd if: \[ f(-x) = -f(x) \]

Take as an example \( f(x) = x^3 \):

• Substituting \( -x \): \( f(-x) = (-x)^3 = -x^3 \)

Here, \( f(-x) = -f(x) \), confirming that \( x^3 \) is an odd function.Odd functions often encompass powers of x raised to odd numbers, such as \( x, x^3, x^5 \). This property makes them visually symmetrical about the origin. Identifying odd functions involves checking whether rotating the graph 180 degrees results in the same image.