Question

A function \(f(x)\) is said to be even if \(f(-x) = f(x)\), and odd if \(f(-x) = -f(x)\). Thus, for example, the function given by \(f(x) = x^2\) is even, while the function given by \(f(x) = x^3\) is odd. Using this definition, answer the following questions.
The sum of two odd functions

• A. is always an even function
• B. is always an odd function
• C. is sometimes odd and sometimes even
• D. may be neither odd nor even

Hint:

Let \(f\) and \(g\) be odd and check \((f + g)(-x)\).

Answer 1

The Correct Option is B

Step 1: Set up:
Let \(f\) and \(g\) be odd, so \(f(-x) = -f(x)\) and \(g(-x) = -g(x)\).

Step 2: Sum:
\((f + g)(-x) = f(-x) + g(-x) = -f(x) - g(x) = -(f + g)(x)\).

Step 3: Conclusion:
The sum is always odd. (It is even only in the trivial case of the zero function, which is also odd.)

Final Answer:
The correct option is (2): is always an odd function.