To find the inverse function $f^{-1}(x)$ for the given function $f(x) = \frac{2x+3}{x-4}$, follow these steps:

Step 1: Set $y = f(x)$

We start by rewriting the function with $y$ in place of $f(x)$: $y = \frac{2x + 3}{x - 4}$

Step 2: Swap $y$ and $x$

To find the inverse, swap $y$ and $x$: $x = \frac{2y + 3}{y - 4}$

Step 3: Solve for $y$

Now, solve this equation for $y$:

1. Multiply both sides by $y - 4$ to eliminate the denominator: $x(y - 4) = 2y + 3$
2. Expand the left side: $xy - 4x = 2y + 3$
3. Move all terms involving $y$ to one side and constants to the other: $xy - 2y = 4x + 3$
4. Factor out $y$ on the left side: $y(x - 2) = 4x + 3$
5. Solve for $y$: $y = \frac{4x + 3}{x - 2}$

Step 4: Write the inverse function

Thus, the inverse function is: $f^{-1}(x) = \frac{4x + 3}{x - 2}$

Would you like more details or have any questions?

Related Questions:

1. How can you verify that a function and its inverse are correct?
2. What is the domain of $f(x)$ and $f^{-1}(x)$?
3. How does the graph of a function compare to the graph of its inverse?
4. Can all functions have an inverse? If not, why?
5. What are the key properties of inverse functions?

Tip:

When finding an inverse function, always check the domain and range to ensure the inverse makes sense in context!