Let's analyze the given expression, $\frac{f(x)}{g(x)} + f(x)g(x)$, assuming that both $f(x)$ and $g(x)$ are odd functions.

1. Properties of odd functions:

A function $h(x)$ is odd if: $h(-x) = -h(x)$ for all $x$. This property holds for both $f(x)$ and $g(x)$ since they are both odd.

2. Analyzing each term in the expression:

a. The quotient $\frac{f(x)}{g(x)}$:

Let's check if this term is odd or even: $\frac{f(-x)}{g(-x)} = \frac{-f(x)}{-g(x)} = \frac{f(x)}{g(x)}$ Since the quotient remains the same when evaluated at $-x$, this term is even.

b. The product $f(x)g(x)$:

Now, consider the product of two odd functions: $f(-x)g(-x) = (-f(x))(-g(x)) = f(x)g(x)$ Again, this expression remains the same when evaluated at $-x$, so this term is also even.

3. Conclusion:

Both terms, $\frac{f(x)}{g(x)}$ and $f(x)g(x)$, are even functions. Therefore, the sum of these two terms: $\frac{f(x)}{g(x)} + f(x)g(x)$ is also an even function.

Would you like further clarification or details on any step?

Here are 5 related questions to expand on this topic:

1. How do we define the difference between odd and even functions?
2. What happens if one function is odd and the other is even in such expressions?
3. Can the product of an odd and an even function be classified as odd or even?
4. How do composite functions behave with odd and even functions?
5. What are examples of real-world applications of odd and even functions?

Tip: In general, the product or quotient of two odd functions results in an even function.