Question

Let \(E\) be an even function and O be an odd function. Determine the symmetry, if any, of the following functions. $$0 \cdot 0$$

Short answer

Answer: The composition \(O \circ O\) has the symmetry of an odd function, which is symmetry about the origin (rotational symmetry of \(180^{\circ}\) about the origin).

Step-by-step solution

Understanding even and odd functions

Even functions are those which satisfy the identity \(E(-x) = E(x)\) for all \(x\). They exhibit symmetry about the y-axis. Odd functions are those which satisfy the identity \(O(-x) = -O(x)\) for all \(x\). They exhibit symmetry about the origin, meaning they are symmetric when rotated by \(180^{\circ}\) about the origin.

Deriving the property of \(O \circ O\)

We have to find out the symmetry of O(O(x)), if there is any. Let's evaluate \((O \circ O)(x)\) and \((O \circ O)(-x)\). First, we will evaluate \((O \circ O)(x)\). We have: $$(O \circ O)(x) = O(O(x))$$ Now, let's evaluate \((O \circ O)(-x)\) by using the property of odd functions: $$(O \circ O)(-x)=O(O(-x))$$ Since \(O\) is an odd function, we can replace \(O(-x)\) by \(-O(x)\): $$O(O(-x))= O(-O(x))$$ Finally, using the odd function property on the composition once again, we get: $$O(-O(x)) = -O(O(x))$$ Thus, we have found that \((O \circ O)(-x) = -O(O(x)) = -(O \circ O)(x)\). This is the property of an odd function.

Concluding the symmetry of \(O \circ O\)

Since \((O \circ O)(-x) = -(O \circ O)(x)\), the overall function \(O \circ O\) has the properties of an odd function. Therefore, the composition \(O \circ O\) exhibits the symmetry of an odd function, which is symmetry about the origin (rotational symmetry of \(180^{\circ}\) about the origin).

Key concepts

Function Symmetry

Functions can display a sense of balance or uniformity that mathematicians refer to as symmetry. In the context of even and odd functions, symmetry plays a special role.
Even functions are characterized by their mirror-like balance across the y-axis. This means if you fold the graph along the y-axis, both halves match perfectly. Mathematically, this translates to the equation: \( E(-x) = E(x) \) for all values of \( x \).

• This type of symmetry is called y-axis symmetry.
• Common examples include the functions \( f(x) = x^2 \) and \( f(x) = \cos(x) \).

Odd functions, on the other hand, exhibit what's called "origin symmetry."
This means if you were to rotate their graph 180 degrees around the origin, the graph would look the same. This rotational symmetry equation is expressed as: \( O(-x) = -O(x) \), again for all \( x \).

• Examples include \( f(x) = x^3 \) and \( f(x) = \sin(x) \).

Composing Functions

Composing functions is a powerful way to combine two or more functions to create a new one with unique properties. This operation is defined as the application of one function to the results of another.
For two functions, \( f \) and \( g \), the composition is represented as \( f(g(x)) \), meaning apply \( g \) first then \( f \) on the result. With odd and even functions, composing can produce interesting results.Given two odd functions \( O \), composing them, as shown in the process of finding \( O(O(x)) \), results in another odd function. This is due to the intrinsic properties of symmetry associated with odd functions:

• When evaluating \( O(O(x)) \), you apply the odd function rule twice.
• This helps deduce that \( (O \circ O)(-x) = -O(O(x)) \), confirming that the composed function retains origin symmetry.

Understanding function composition is crucial for solving complex mathematical problems where combining simpler functions creates more elaborate structures.

Odd Function Properties

Odd functions possess distinct traits tied to their symmetry about the origin. Recognizing these properties can simplify analyzing graphs and understanding complex relationships within functions. The primary property defining odd functions is:For any odd function \( O \), \[ O(-x) = -O(x) \]This means the value of the function at negative inputs is the negative of the value at positive inputs.

• This leads to a graph that when rotated 180° about the origin, remains unchanged.

One useful aspect of odd functions is how they respond to function operations such as composition:

• Composing two odd functions as \( O(O(x)) \) results in another odd function, maintaining symmetry.
• This property is a useful tool to predict and verify symmetry in complex function analyses.

Grasping these properties helps in understanding transformations in function behavior when expressed in different mathematical forms or when evaluating their integrals.