Question

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

Short answer

Answer: No, the function has no symmetry since it is neither even nor odd.

Step-by-step solution

Recall the properties of even and odd functions

An even function is a function with the property \(f(-x) = f(x)\) for all \(x\) in its domain. An odd function is a function with the property \(f(-x) = -f(x)\) for all \(x\) in its domain.

Determine the symmetry of the given function

Let the given function be denoted as \(G(x)=\frac{E(x)}{O(x)}\). We will now determine the symmetry by finding the value of \(G(-x)\) and comparing it to \(G(x)\).

Compute \(G(-x)\)

Substitute \(-x\) in place of \(x\) in the given function: $$G(-x)=\frac{E(-x)}{O(-x)}$$

Apply properties of even and odd functions

Since \(E\) is an even function, \(E(-x) = E(x)\). Similarly, since \(O\) is an odd function, \(O(-x) = -O(x)\). Therefore: $$G(-x)=\frac{E(x)}{-O(x)}$$

Compare \(G(-x)\) and \(G(x)\)

Comparing the expressions for \(G(-x)\) and \(G(x)\), we see that they are not equal: $$G(-x) \neq G(x)$$ which means the function is not even. Also, \(G(-x)\) is not the negative of \(G(x)\): $$G(-x) \neq -G(x)$$ which means the function is not odd. Thus, the function \(\frac{E}{O}\) has no symmetry since it is neither even nor odd.

Key concepts

Symmetry in Mathematics

Understanding symmetry is a fundamental aspect of mathematics. Symmetry refers to a situation where one part of an object or system mirrors another. In the context of functions, symmetry can tell us a lot about how a function behaves.

There are two primary types of symmetry associated with functions: even symmetry and odd symmetry. These are determined by how the function behaves when the input values are negated.

• **Even functions**: A function is said to have even symmetry if it satisfies the condition \( f(-x) = f(x) \) for every \( x \) in its domain. This means the function's graph is mirrored along the y-axis.
• **Odd functions**: A function is called odd if it satisfies \( f(-x) = -f(x) \), meaning its graph exhibits rotational symmetry around the origin.

However, not all functions have even or odd symmetry. Some may display neither, as demonstrated with the combination of an even and an odd function, \( \frac{E}{O} \), where symmetry is not guaranteed.

Properties of Functions

The properties of functions such as evenness and oddness play a crucial role in determining function behavior. For even and odd functions, these properties dictate how the function's values are related to those at negative inputs.

Understanding these properties helps in solving problems and simplifying expressions in mathematics.

• **Even functions** maintain their value when the sign of the input is changed, making them symmetric along the y-axis.
• **Odd functions** change the sign of their output when the input sign is changed, indicating a type of rotational symmetry.

Applying the properties of these functions can be useful in various mathematical operations, such as integration, where the irregularity of odd functions about specific bounds allows certain simplifications.

Function Composition

Function composition is a process of applying one function to the results of another. This technique is fundamental, allowing us to construct complex behaviors from simpler, well-understood ones.

In general, composing functions involves substituting the entire formula of one function as the input of another.

• For instance, if we have two functions, \( f(x) \) and \( g(x) \), the composition would be expressed as \( f(g(x)) \).
• Function composition is associative, which means \( f(g(h(x))) \) is the same as \( f((g(h(x)))) \).

When combining an even function with an odd one, or evaluating such compositions, examining the symmetry of the resulting function involves applying the properties of each individual function's type. However, as noted, \( \frac{E}{O} \) lacks symmetry indicating that function composition doesn't always yield results that maintain original symmetries.