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: The function E+O has no specific symmetry and is neither even nor odd.

Step-by-step solution

Define Even and Odd Functions

We should first recall the definition of even and odd functions. A function E(x) is called even if E(-x) = E(x) for all x in its domain. Similarly, a function O(x) is called odd if O(-x) = -O(x) for all x in its domain.

Analyze the Sum of Even and Odd functions

Now, let's analyze the function (E+O)(x). We will evaluate it at the points (-x) and x and compare their values. (E+O)(-x) = E(-x) + O(-x) Since E is an even function: E(-x) = E(x) And since O is an odd function: O(-x) = -O(x) So, (E+O)(-x) = E(x) - O(x)

Check if E+O is Even or Odd

Now, let's check if (E+O)(-x) is equal to (E+O)(x) or -(E+O)(x) (E+O)(x) = E(x) + O(x) Compare (E+O)(-x) and (E+O)(x) : (E+O)(-x) = E(x) - O(x) (E+O)(x) = E(x) + O(x) Since (E+O)(-x) is not equal to (E+O)(x), the function E+O is not an even function. Now, let's compare (E+O)(-x) and -(E+O)(x): (E+O)(-x) = E(x) - O(x) -(E+O)(x) = -E(x) - O(x) Since (E+O)(-x) is not equal to -(E+O)(x), the function E+O is not an odd function.

Conclusion

The sum of the even function E and the odd function O, i.e., the function E+O, has no specific symmetry and is neither even nor odd.

Key concepts

Symmetry of Functions

Functions often exhibit special properties like symmetry that can help us understand their behavior. Symmetry in functions means that the graph of the function can be reflected over a line or rotated 180 degrees.

There are two common types of symmetry:

• Reflectional Symmetry: A graph has reflectional symmetry about the y-axis if the left side is a mirror image of the right side. This is typical for even functions.
• Rotational Symmetry: If you can rotate the graph 180 degrees around the origin and it looks the same, the function has rotational symmetry, which is characteristic of odd functions.

Understanding these symmetries helps to predict how combining different types of functions will behave. They reveal much about the function's nature without needing to plot every detail.

Definition of Even Functions

An even function is one that satisfies the condition: for any value of x within the domain, if you replace x with -x, the function's value remains unchanged. Formally, this can be stated as:
\[ E(x) = E(-x) \]

Features of even functions include:

• They exhibit reflectional symmetry across the y-axis.
• Examples of even functions are those like the quadratic function \( f(x) = x^2 \) and the cosine function \( f(x) = \cos x \).

Identifying a function as even can simplify calculations and provide insights into its graphical representation.

Definition of Odd Functions

Odd functions have a unique property wherein substituting -x for x changes the sign of the function value. The mathematical expression for this is:
\[ O(x) = -O(-x) \]

Characteristics of odd functions include:

• They possess rotational symmetry about the origin, meaning the graph looks the same when rotated 180 degrees around the origin.
• Examples of odd functions include linear functions like \( f(x) = x \) and trigonometric functions such as \( f(x) = \sin x \).

Recognizing a function as odd provides a powerful tool for understanding its symmetry and behavior, aiding in problem-solving and analysis.