Find the inverse of the function f(x)=16 - (x-5)^2 / 4

To find the inverse of the function \( f(x) = 16 - \frac{(x-5)^2}{4} \), follow these steps:

1. **Rewrite the function with \( y \) in place of \( f(x) \):**
   \[
   y = 16 - \frac{(x-5)^2}{4}
   \]

2. **Solve for \( x \) in terms of \( y \):**

   \[
   y = 16 - \frac{(x-5)^2}{4}
   \]

   Subtract 16 from both sides:

   \[
   y - 16 = -\frac{(x-5)^2}{4}
   \]

   Multiply both sides by -4:

   \[
   -4(y - 16) = (x-5)^2
   \]

   Simplify the left side:

   \[
   4(16 - y) = (x-5)^2
   \]

   Take the square root of both sides:

   \[
   \sqrt{4(16 - y)} = x - 5
   \]

   \[
   2\sqrt{16 - y} = x - 5
   \]

   Add 5 to both sides:

   \[
   x = 5 + 2\sqrt{16 - y}
   \]

3. **Interchange \( x \) and \( y \) to write the inverse function:**

   \[
   y = 5 + 2\sqrt{16 - x}
   \]

4. **Write the final form of the inverse function:**

   \[
   f^{-1}(x) = 5 + 2\sqrt{16 - x}
   \]

The inverse function of \( f(x) = 16 - \frac{(x-5)^2}{4} \) is:

\[
f^{-1}(x) = 5 + 2\sqrt{16 - x}
\]

Do you want details or have any questions?

Here are 5 numbered questions you might find interesting:
1. How do you find the derivative of an inverse function?
2. What is the domain and range of the original and inverse function?
3. How do you verify if two functions are inverses of each other?
4. Can you find the inverse of a function if it is not one-to-one?
5. What are the applications of inverse functions in real-life scenarios?

**Tip:** When finding the inverse of a function, always check the domain of the original function to ensure the inverse function is properly defined within its domain.

Discover how to find the inverse of the function f(x) = 16 - (x-5)^2 / 4 step-by-step. This problem involves inverse functions and quadratic equations, essential concepts in advanced high school mathematics.

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