We are given the function $f(x) = \sqrt{4x - 9}$, and we are tasked with finding its inverse $f^{-1}(x)$.

Steps to find the inverse function:

1. Write the equation:
   $y = \sqrt{4x - 9}$ where $y = f(x)$.

2. Swap $x$ and $y$:
   To find the inverse, we swap $x$ and $y$ and solve for $y$: $x = \sqrt{4y - 9}$

3. Square both sides:
   This removes the square root on the right-hand side: $x^2 = 4y - 9$

4. Solve for $y$:
   $4y = x^2 + 9$ $y = \frac{x^2 + 9}{4}$

Thus, the inverse function is: $f^{-1}(x) = \frac{x^2 + 9}{4}$

Final Answer:

$f^{-1}(x) = \frac{x^2 + 9}{4}$

Would you like more details or have any other questions on this topic?

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Follow-up questions:

1. How would you find the domain and range of the original and inverse functions?
2. Can we graph both $f(x)$ and $f^{-1}(x)$ on the same coordinate plane?
3. What is the general method for finding inverses of functions involving square roots?
4. How does the concept of a function's inverse relate to the concept of symmetry?
5. How would the inverse change if the square root had a different form, like $\sqrt{2x - 5}$?

Tip: When finding inverses, always swap the $x$ and $y$ values, then solve for $y$.