The given function is $f(x) = \frac{3x + 4}{x}$. To find its inverse, $f^{-1}(x)$, we need to follow these steps:

1. Rewrite $f(x)$ as $y$: $y = \frac{3x + 4}{x}$

2. Solve for $x$ in terms of $y$: Multiply both sides of the equation by $x$ to eliminate the denominator: $yx = 3x + 4$ Now, isolate terms involving $x$: $yx - 3x = 4$ Factor out $x$: $x(y - 3) = 4$ Solve for $x$: $x = \frac{4}{y - 3}$

3. Replace $y$ with $x$ to express the inverse function: $f^{-1}(x) = \frac{4}{x - 3}$

Thus, the inverse of $f(x)$ is $f^{-1}(x) = \frac{4}{x - 3}$.

The correct option in the image is the third option: $f^{-1}(x) = \frac{4}{x - 3}$.

Would you like more details on the steps, or do you have any questions?

Here are 5 related questions to explore further:

1. What is the general method for finding the inverse of a rational function?
2. How can you check if a function's inverse exists?
3. Can all rational functions have an inverse?
4. What are the conditions under which a function is one-to-one and thus has an inverse?
5. How do you verify if two functions are inverses of each other?

Tip: When solving for the inverse, always ensure that the function is one-to-one (passes the horizontal line test) before proceeding.