To find the function $\left(\frac{f}{g}\right)(x)$, you need to divide the two given functions:

$\left(\frac{f}{g}\right)(x) = \frac{\frac{3x+1}{x^2-25}}{\frac{2x-4}{x^2-25}}$

When dividing fractions, you multiply the numerator by the reciprocal of the denominator. So, this becomes:

$\left(\frac{f}{g}\right)(x) = \frac{3x+1}{x^2-25} \times \frac{x^2-25}{2x-4}$

Since the $x^2 - 25$ terms in the numerator and denominator are the same, they cancel out, simplifying to:

$\left(\frac{f}{g}\right)(x) = \frac{3x + 1}{2x - 4}$

This is the simplified form of the function $\left(\frac{f}{g}\right)(x)$.

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Related Questions:

1. What are the domain restrictions for $\left(\frac{f}{g}\right)(x)$?
2. How do you factor the expression $2x - 4$?
3. Can $\left(\frac{f}{g}\right)(x)$ be simplified further?
4. How would you find the roots of the numerator and denominator separately?
5. How would you find the inverse of $\left(\frac{f}{g}\right)(x)$?

Tip: Always check the domain restrictions after simplifying, especially in rational functions, to avoid division by zero.