The problem requires finding the inverse of the function $f(x) = x^2 - 4$ and then simplifying the answer.

Step 1: Write the function in terms of $y$.

$y = x^2 - 4$

Step 2: Swap $x$ and $y$ to find the inverse.

$x = y^2 - 4$

Step 3: Solve for $y$ in terms of $x$.

$y^2 = x + 4$ $y = \pm \sqrt{x + 4}$

Step 4: Identify the correct inverse function.

The inverse function $f^{-1}(x)$ is not a single function because it depends on the domain of the original function $f(x)$. The original function $f(x) = x^2 - 4$ is not one-to-one over all real numbers because it is a parabola. Typically, to have an inverse, the function is restricted to $x \geq 0$ (for the principal square root) or $x \leq 0$.

Thus, the inverse functions could be: $f^{-1}(x) = \sqrt{x + 4} \quad \text{for} \quad x \geq -4$ or $f^{-1}(x) = -\sqrt{x + 4} \quad \text{for} \quad x \geq -4$

Simplified Answer:

The inverse function is $f^{-1}(x) = \sqrt{x + 4}$ or $f^{-1}(x) = -\sqrt{x + 4}$.

Would you like to see the graph of the inverse functions or need further details?

Here are some related questions you might consider:

1. How do we determine the domain and range of $f(x) = x^2 - 4$?
2. What is the significance of restricting the domain for finding the inverse of a function?
3. How can we verify that the inverse function is correct?
4. How do we handle inverses for more complex functions like cubic or rational functions?
5. What are the graphical implications of the inverse function on the coordinate plane?

Tip: Always check the domain of the original function before finding the inverse to ensure that the inverse function is valid.