Problem 75 - Algebra and Trigonometry

Question

Sums of Even and Odd Functions If \(f\) and \(g\) are both even functions, is \(f+g\) necessarily even? If both are odd, is their sum necessarily odd? What can you say about the sum if one is odd and one is even? In each case, prove your answer.

Step-by-Step Solution

Answer

The sum of two even functions is even, the sum of two odd functions is odd, and the sum of one even and one odd function is neither even nor odd.

Step 1: Understanding Even and Odd Functions

An even function satisfies the condition \( f(x) = f(-x) \) for all \( x \). In contrast, an odd function satisfies \( g(x) = -g(-x) \). These definitions are key to understanding the behavior of their sums.

Step 2: Case 1: Sum of Two Even Functions

Let \( f(x) \) and \( g(x) \) be even functions. Consider their sum, \( h(x) = f(x) + g(x) \). For \( h \) to be even, we need \( h(x) = h(-x) \). Substitute to get \( h(-x) = f(-x) + g(-x) \). Since \( f \) and \( g \) are even, \( f(-x) = f(x) \) and \( g(-x) = g(x) \), leading to \( h(-x) = f(x) + g(x) = h(x) \). Hence, \( h \) is even.

Step 3: Case 2: Sum of Two Odd Functions

Let \( f(x) \) and \( g(x) \) be odd functions. Consider their sum, \( h(x) = f(x) + g(x) \). For \( h \) to be odd, we need \( h(x) = -h(-x) \). Substitute to get \( h(-x) = f(-x) + g(-x) \). Since \( f \) and \( g \) are odd, \( f(-x) = -f(x) \) and \( g(-x) = -g(x) \), leading to \( h(-x) = -f(x) - g(x) = -(f(x) + g(x)) = -h(x) \). Hence, \( h \) is odd.

Step 4: Case 3: Sum of One Even and One Odd Function

Let \( f(x) \) be even and \( g(x) \) be odd. Consider their sum, \( h(x) = f(x) + g(x) \). Evaluate \( h(-x) = f(-x) + g(-x) \). Using the properties, \( f(-x) = f(x) \) and \( g(-x) = -g(x) \). Therefore, \( h(-x) = f(x) - g(x) \), which is neither equal to \( h(x) \) for even, nor \( -h(x) \) for odd. Thus, \( h \) is neither even nor odd.

Key Concepts

Sum of Even Functions

When you have two even functions, their sum will also be an even function. Remember, an even function is one where the equation satisfies the condition \( f(x) = f(-x) \) for every value of \( x \). If we take two even functions, \( f(x) \) and \( g(x) \), the sum \( h(x) = f(x) + g(x) \) needs to meet the condition \( h(x) = h(-x) \) to be even.

Let's check: compute \( h(-x) \), which is \( f(-x) + g(-x) \). Since both \( f \) and \( g \) are even, that means \( f(-x) = f(x) \) and \( g(-x) = g(x) \). Substitute these values in and you get \( h(-x) = f(x) + g(x) = h(x) \).

Therefore, the sum of two even functions is again an even function, as it holds true to the definition of being even.

Sum of Odd Functions

When adding two odd functions, the result is always an odd function. An "odd" function is defined by the relationship \( g(x) = -g(-x) \). To explore this, consider two odd functions, \( f(x) \) and \( g(x) \). If we sum them, \( h(x) = f(x) + g(x) \), we need \( h(x) = -h(-x) \) to confirm it's odd.

Let's do the check: calculate \( h(-x) = f(-x) + g(-x) \). Knowing that \( f(x) \) and \( g(x) \) are odd, we have \( f(-x) = -f(x) \) and \( g(-x) = -g(x) \). Substitution back gives us \( h(-x) = -f(x) - g(x) = -(f(x) + g(x)) = -h(x) \).

This confirms that the sum of two odd functions is indeed an odd function, satisfying the necessary properties.

Properties of Even and Odd Functions

Understanding the properties of even and odd functions is crucial, especially when combining them. Let's break down their interactions:

• Sum of an Even and Odd Function: When you add an even function to an odd function, the resulting function is neither even nor odd. This is because an even function requires \( h(x) = h(-x) \), whereas an odd function requires \( h(x) = -h(-x) \). The combined function will fulfill neither of these properties. This is shown when \( h(x) = f(x) + g(x) \) leads to \( h(-x) = f(x) - g(x) \), neither matching \( h(x) \) nor \( -h(x) \).
• Subtracting Functions: Subtracting one function from another can affect whether the result remains even, odd, or becomes neither. If you subtract an even function from another even function, the result is still even. Similarly, subtracting two odd functions results in an odd function.
• Multiplication of Even and Odd Functions: If you multiply even functions together, the result is even. However, if you multiply two odd functions, the result is even. Multiplying an even function by an odd function results in an odd function.

These rules help in predicting how the operations on functions will affect their evenness or oddness, thus aiding in understanding and simplifying complex problems.