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The given function is:

\[
f(x) = \frac{2x - 1}{3x + 4}
\]

with the condition \( x \neq -\frac{4}{3} \).

We are tasked with finding the inverse function \( f^{-1}(x) \).

### Steps to find the inverse function:

1. **Replace \( f(x) \) with \( y \)**:
   \[
   y = \frac{2x - 1}{3x + 4}
   \]
   
2. **Switch \( x \) and \( y \)** to find the inverse:
   \[
   x = \frac{2y - 1}{3y + 4}
   \]

3. **Solve for \( y \)** (the inverse function):
   - Multiply both sides by \( 3y + 4 \):
     \[
     x(3y + 4) = 2y - 1
     \]
   - Expand:
     \[
     3xy + 4x = 2y - 1
     \]
   - Collect terms involving \( y \) on one side:
     \[
     3xy - 2y = -1 - 4x
     \]
   - Factor out \( y \) on the left-hand side:
     \[
     y(3x - 2) = -1 - 4x
     \]
   - Solve for \( y \):
     \[
     y = \frac{-1 - 4x}{3x - 2}
     \]

Thus, the inverse function is:

\[
f^{-1}(x) = \frac{-1 - 4x}{3x - 2}
\]

Would you like a detailed explanation or have any further questions?

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Here are five related questions to explore:

1. How do you verify that a function and its inverse are correct?
2. What are the conditions for a function to have an inverse?
3. How do you graph a function and its inverse on the same plane?
4. What does the inverse of a function represent geometrically?
5. How does the domain and range of a function change with its inverse?

**Tip:** Always remember that a function must be one-to-one (bijective) to have an inverse, meaning it passes both the vertical and horizontal line tests.

Fungsi f: R → R didefinisikan sebagai f(x) = (2x - 1) / (3x + 4), dengan x ≠ -4/3. Invers dari fungsi f adalah f^(-1)(x) = ....

Learn how to find the inverse of the function f(x) = (2x - 1) / (3x + 4) where x ≠ -4/3. This detailed solution covers key algebraic concepts including rational functions and the process to derive inverse functions. Suitable for high school students in Grades 10-12.

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