https://file.aiquickdraw.com/mathbot/file/1726900985666p6xsryj2.jpg

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.

Which of the following is the inverse of f(x) = (3x + 4) / x, if it exists?

Learn how to find the inverse of the rational function f(x) = (3x + 4) / x with a step-by-step solution. Explore key concepts in inverse functions and algebra, suitable for high school students in Grades 10-12.

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