To find the inverse of the function $y = \frac{1}{4}x^2 - 4$, follow these steps:

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

Let $y = \frac{1}{4}x^2 - 4$.

2. Swap $x$ and $y$:

To find the inverse, swap $x$ and $y$: $x = \frac{1}{4}y^2 - 4$

3. Solve for $y$:

Now, solve for $y$ in terms of $x$.

• Add 4 to both sides: $x + 4 = \frac{1}{4}y^2$

• Multiply both sides by 4 to eliminate the fraction: $4(x + 4) = y^2$

• Take the square root of both sides: $y = \pm \sqrt{4(x + 4)}$

Since square roots yield two values, you need to consider the domain and range of the original function to determine which branch of the square root applies. The original function $y = \frac{1}{4}x^2 - 4$ is a parabola that opens upwards. For the inverse to be a function, we typically take only the positive square root.

Thus, the inverse is: $y^{-1}(x) = 2\sqrt{x + 4}$

Final Answer:

$y^{-1}(x) = 2\sqrt{x + 4}$

Let me know if you want more details or have any questions! Here are some related questions:

1. How do you find the domain and range of the inverse function?
2. What is the graphical interpretation of the inverse function?
3. Can the negative square root be part of the inverse?
4. How do you check if the inverse function is correct?
5. How do you differentiate an inverse function?

Tip: To verify that a function is the inverse of another, compose the two functions (i.e., $f(f^{-1}(x))$ and $f^{-1}(f(x))$) and check if both yield $x$.