Let's consider the definitions of odd and even functions:

• A function $f$ is odd if for all $x \in \mathbb{R}$, $f(-x) = -f(x)$.
• A function $f$ is even if for all $x \in \mathbb{R}$, $f(-x) = f(x)$.

Given that both $f$ and $g$ are odd functions, we have:

• $f(-x) = -f(x)$ for all $x$.
• $g(-x) = -g(x)$ for all $x$.

Now, we need to determine whether $f - g$ is odd or even. Let's check the behavior of $f - g$ when $-x$ is substituted:

$(f - g)(-x) = f(-x) - g(-x)$

Using the fact that $f$ and $g$ are odd:

$(f - g)(-x) = -f(x) - (-g(x)) = -f(x) + g(x) = -(f(x) - g(x)) = -(f - g)(x)$

Thus, we see that:

$(f - g)(-x) = -(f - g)(x)$

This shows that $f - g$ is odd.

Conclusion:

The function $f - g$ is odd.

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Would you like further details or clarifications on the properties of odd and even functions? Here are some related questions to explore:

1. What is the sum of two odd functions? Is the result odd or even?
2. How do odd and even functions interact with multiplication? (e.g., is the product of two odd functions odd or even?)
3. How can you prove that a function is even or odd by using symmetry?
4. What happens if you combine an odd function with an even function? Is the result odd, even, or neither?
5. Can you give an example of a function that is neither odd nor even?

Tip: The key to recognizing whether a function is odd or even often lies in checking how it behaves under $x \to -x$ transformations!