To find $f^{-1}(-4)$, we need to first determine the inverse function $f^{-1}(x)$ of the given function $f(x) = \sqrt{2x} - 6$. Here are the steps to do this:

Step 1: Set up the equation for the inverse

We start by setting $y = f(x) = \sqrt{2x} - 6$ and then solve for $x$ in terms of $y$.

$y = \sqrt{2x} - 6$

Step 2: Isolate the square root term

Add 6 to both sides of the equation:

$y + 6 = \sqrt{2x}$

Step 3: Square both sides to remove the square root

Square both sides:

$(y + 6)^2 = 2x$

Step 4: Solve for $x$

Now, divide by 2 to isolate $x$:

$x = \frac{(y + 6)^2}{2}$

Thus, the inverse function is:

$f^{-1}(x) = \frac{(x + 6)^2}{2}$

Step 5: Evaluate $f^{-1}(-4)$

Now, substitute $x = -4$ into the inverse function:

$f^{-1}(-4) = \frac{(-4 + 6)^2}{2} = \frac{(2)^2}{2} = \frac{4}{2} = 2$

Final Answer:

$f^{-1}(-4) = 2$

Would you like a detailed explanation of any part of the solution? Here are some related questions to consider:

1. How do you verify that $f^{-1}(x)$ is the correct inverse function?
2. Can a function always have an inverse? What conditions must be met?
3. How do you graphically interpret inverse functions?
4. How do transformations affect the function and its inverse?
5. What happens to the domain and range of a function and its inverse?

Tip: To find the inverse of a function, always swap $x$ and $y$ and solve for the new $y$, which represents $f^{-1}(x)$.