Yes, if $f$ and $g$ are both odd functions, then $f + g$ is also an odd function.

Reason:

A function $f(x)$ is considered odd if it satisfies the property: $f(-x) = -f(x) \quad \text{for all} \, x.$

Similarly, for $g(x)$, if it is odd: $g(-x) = -g(x) \quad \text{for all} \, x.$

Now, consider the sum of these two functions $f(x) + g(x)$. We check whether it satisfies the condition of being odd: $(f + g)(-x) = f(-x) + g(-x).$ Since both $f$ and $g$ are odd, we have: $f(-x) = -f(x) \quad \text{and} \quad g(-x) = -g(x).$ Thus: $(f + g)(-x) = -f(x) + (-g(x)) = -(f(x) + g(x)).$ This shows that: $(f + g)(-x) = -(f + g)(x),$ which is the definition of an odd function.

Conclusion:

Since $f + g$ satisfies the condition for being odd, the sum of two odd functions is indeed odd.

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Would you like a more detailed explanation or any specific examples? Here are some related questions you might find interesting:

1. What happens if you add an even function to an odd function?
2. Can a function be both even and odd? If so, what are the conditions?
3. How do we prove that the product of two odd functions is odd?
4. What is the graphical interpretation of an odd function?
5. How do we differentiate odd and even functions algebraically?

Tip: When working with odd and even functions, always remember that an odd function has symmetry about the origin, while an even function has symmetry about the y-axis.