Question

Let \(E\) be an even function and O be an odd function. Determine the symmetry, if any, of the following functions. $$E \cdot O$$

Short answer

Answer: The product has odd symmetry.

Step-by-step solution

Define Even and Odd Functions

An even function primarily satisfies the following condition: for all \(x\) in the domain of \(E\), \(E(-x)=E(x)\). Examples of even functions include the functions \(f(x) = x^2\) and \(f(x) = 1\). An odd function, however, satisfies the condition: for all \(x\) in its domain, \(O(-x)=-O(x)\). Examples of odd functions are \(f(x) = x^3\) and \(f(x) = x\).

Analyze the Product of the Given Functions

We have a product of an even function and an odd function. Let's denote the product as a new function \(F(x) = E(x) \cdot O(x)\). Now we need to find the symmetry of \(F(x)\) in terms of even or odd function property, if any.

Determine the Symmetry of the Product Function

To determine whether the function \(F(x) = E(x) \cdot O(x)\) is even, odd, or has no symmetry, we will look at the value of \(F(-x)\) and compare it to \(F(x)\). Let's find \(F(-x)\): $$F(-x) = E(-x)\cdot O(-x)$$ Using the known properties of even and odd functions, substitute the relevant values: $$F(-x) = E(x)\cdot (-O(x)) = -E(x)O(x)$$ We can see that \(F(-x) = -F(x)\). This shows that the product of an even function and an odd function is an odd function. The symmetry of \(F(x)\) would be odd symmetry.

Key concepts

Even Functions

An even function has a distinct property that defines its symmetry. The key condition for an even function is that it fulfills the equation:

• \(E(-x) = E(x)\)

This means that if you take any value from the domain and invert the sign of the input, the output remains unchanged.
Imagine the graph of an even function as having a perfect mirror-image symmetry along the y-axis. Common examples include basic functions like:

• \(f(x) = x^2\)
• \(f(x) = \cos x\)

These functions demonstrate that for every positive input, the function behaves the same as it would if the input were negative.This property is not limited only to polynomial functions. It applies across any function where this defining condition holds true.

Odd Functions

Odd functions possess a unique symmetry around the origin. For a function to be classified as odd, it must satisfy:

• \(O(-x) = -O(x)\)

This relationship indicates a 180-degree rotational symmetry about the origin on the graph.
Visualize the curve: if one part of it lies in the first quadrant, its corresponding part would be found inverted in the third quadrant. Classic odd function examples are:

• \(f(x) = x^3\)
• \(f(x) = \sin x\)

In practical terms, the odd symmetry implies that the function essentially mimics itself over a full rotation around the origin. This results in values being negated when their inputs are negated.

Function Product Symmetry

When considering the product of two functions, symmetry often becomes less intuitive. For instance, multiplying an even function \(E\) and an odd function \(O\), we investigate the resulting function's symmetry.The product function, denoted as \(F(x) = E(x)\cdot O(x)\), requires checking the symmetry properties by evaluating \(F(-x)\):

• \(F(-x) = E(-x)\cdot O(-x)\)
• Substituting the symmetry properties, we find \(F(-x) = E(x)\cdot (-O(x)) = -E(x)O(x)\)

Therefore, \(F(-x) = -F(x)\), confirming that the product \(F(x)\) inherits the symmetry of an odd function.
This example illustrates a vital rule in function operations: the product of an even and odd function is always odd. Understanding this outcome can simplify complex function analysis, revealing relationships and symmetries where they might not have been initially apparent.