Question

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

Short answer

Answer: The composite function of two even functions is even.

Step-by-step solution

Define the even and odd functions

Even functions have the property that \(E(-x) = E(x)\), and odd functions have the property that \(O(-x) = -O(x)\).

Write the composition of even functions

We are given the composite function \(E \circ E\). For any input \(x\), the composite function is defined as \((E \circ E)(x) = E(E(x))\).

Analyze the behavior of the composite function with negative input

To determine the symmetry of \((E \circ E)(x)\), we have to check the behavior of the function when evaluated with a negative input - whether it’s even, odd or neither. We do this by analyzing \((E \circ E)(-x)\).

Substitute the property of the even function

We now input \(-x\) into the composite function and let \(E\) act on it. Using the property of the even function (i.e., \(E(-x) = E(x)\)), we can write: $$(E \circ E)(-x) = E(E(-x)) = E(E(x))$$

Determine the symmetry of the composite function

Since, \((E \circ E)(-x) = E(E(x)) = (E \circ E)(x)\), we can see that the composite function maintains the property of an even function. Therefore, the composite function \(E \circ E\) is an even function, and it has symmetry about the \(y\)-axis.

Key concepts

Even and Odd Functions

Even and odd functions are two fundamental types of symmetry in the realm of mathematics. An even function is one that satisfies the condition \( E(-x) = E(x) \), meaning when you input the negative of a number, you get the same output as if you had input the number itself. This results in a graph that is symmetrical about the vertical axis, also known as the y-axis. Think of mirror images: if the right side of the graph is the same as the left when reflected across the y-axis, the function is even.

Odd functions, on the other hand, follow the rule \( O(-x) = -O(x) \). When you input the negative of a number, the output is the negative of what it would be for the positive input. This causes the graph to have rotational symmetry about the origin, which means if you were to rotate the graph 180 degrees around the origin, it would look the same.

Composite Function Analysis

The composition of functions is a concept where you take two functions and combine them into one operation. In mathematical terms, the composition of functions \( g \circ f \) (read as 'g composed with f') is defined as \( (g \circ f)(x) = g(f(x)) \). When you're given two functions and asked to determine the symmetry of their composite, you must look at how the outputs of these functions interact when they're nested inside each other. If both functions are even or both are odd, specific patterns emerge that help us predict the composite function's behavior. It’s like putting one symmetry inside another, and investigating how the resulting combination behaves.

Function Symmetry

Understanding function symmetry is crucial in determining the behavior of composites. Symmetry in functions essentially tells us about the balance and regularity in the shape of the graph. There are different types of symmetry: even functions display y-axis symmetry, odd functions have origin symmetry, and functions that are neither may not display any symmetry. To analyze composite functions, we look at whether their individual parts respect these symmetries. When exploring composites, we plug in negative inputs to test how the output relates to positive inputs. This step is vital, as it essentially 'tests' the function's symmetry type.

Even Function Properties

Focusing on even functions, their properties are not just interesting but hugely useful in mathematical analysis. Beyond the basic definition \( E(-x) = E(x) \), even functions have other intriguing properties. For instance, when you integrate an even function over a symmetric interval like \([-a, a]\), you can double the integral from \([0, a]\) to find the total area under the curve. This is due to the symmetry about the y-axis. Another property is when two even functions are composed, as seen in our exercise with \( E \circ E \), the result is also an even function. This can greatly simplify the process of evaluating or integrating complex expressions.