If f(x) and g(x) are odd functions of x, then which of the following is true?

Options

A. `f+g` is odd, fg is odd
B. `f+g` is odd, fg is even
C. `f+g` is even, fg is odd
D. `f+g` is even, fg is even

Answer

To determine the properties of the functions \( f(x) \) and \( g(x) \) given that they are both odd functions, we can follow these steps:

### Step 1: Understanding Odd Functions
An odd function satisfies the condition:
\[ f(-x) = -f(x) \]
for all \( x \). Similarly, for \( g(x) \):
\[ g(-x) = -g(x) \]

### Step 2: Analyzing \( f(x) + g(x) \)
To analyze \( f(x) + g(x) \), we will substitute \(-x\) into the expression:
\[ f(-x) + g(-x) = -f(x) - g(x) = -(f(x) + g(x)) \]
This shows that \( f(x) + g(x) \) is also an odd function.

### Step 3: Analyzing \( f(x) \cdot g(x) \)
Now we will analyze the product \( f(x) \cdot g(x) \):
\[ f(-x) \cdot g(-x) = (-f(x)) \cdot (-g(x)) = f(x) \cdot g(x) \]
This indicates that \( f(x) \cdot g(x) \) is an even function.

### Conclusion
From the above analysis, we conclude:
- \( f(x) + g(x) \) is an odd function.
- \( f(x) \cdot g(x) \) is an even function.

Thus, the correct statements are:
- \( f(x) + g(x) \) is odd.
- \( f(x) \cdot g(x) \) is even.