You want to find the inverse of the function $f(x) = \sqrt{2x + 3}$. Here's the step-by-step process to solve for $f^{-1}(x)$:

Step 1: Replace $f(x)$ with $y$

$y = \sqrt{2x + 3}$

Step 2: Swap $x$ and $y$ to start solving for the inverse

$x = \sqrt{2y + 3}$

Step 3: Square both sides to remove the square root

$x^2 = 2y + 3$

Step 4: Solve for $y$

$x^2 - 3 = 2y$ $y = \frac{x^2 - 3}{2}$

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

Final Answer:

$f^{-1}(x) = \frac{x^2 - 3}{2}$

Would you like any further details or explanations on this?

Here are 5 questions related to this process:

1. What are the steps to verify if the inverse function is correct?
2. How do domain restrictions affect the inverse of a function like this?
3. What is the domain of the original function, $f(x) = \sqrt{2x + 3}$?
4. How would the graph of the inverse function compare to the original function?
5. Can you find the inverse of other types of radical functions?

Tip: Always check that the domain and range of a function and its inverse align correctly!