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The problem provided asks about the inverse of the function \( f(x) = x^2 - 25 \) on the interval \( [0, \infty) \).

### Part A: Finding the Inverse of \( f(x) \)

To find the inverse of \( f(x) \), follow these steps:
1. **Set \( f(x) = y \):**
   \[
   y = x^2 - 25
   \]
2. **Solve for \( x \) in terms of \( y \):**
   \[
   y + 25 = x^2
   \]
   \[
   x = \sqrt{y + 25}
   \]
   Since the interval is \( [0, \infty) \), we only consider the positive square root.

Thus, the inverse function is:
\[
f^{-1}(x) = \sqrt{x + 25}
\]

### Part B: Compute \( f(f^{-1}(x)) \)

Now, let's compute \( f(f^{-1}(x)) \). Using the function \( f(x) = x^2 - 25 \) and the inverse \( f^{-1}(x) = \sqrt{x + 25} \), we substitute \( f^{-1}(x) \) into \( f(x) \):

1. Start with the function \( f(f^{-1}(x)) \):
   \[
   f(f^{-1}(x)) = f(\sqrt{x + 25})
   \]
2. Apply the function \( f(x) = x^2 - 25 \):
   \[
   f(\sqrt{x + 25}) = (\sqrt{x + 25})^2 - 25
   \]
3. Simplify:
   \[
   f(\sqrt{x + 25}) = (x + 25) - 25 = x
   \]

Thus, \( f(f^{-1}(x)) = x \), which confirms that applying \( f \) to \( f^{-1}(x) \) gives \( x \), as expected.

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Would you like more details or explanations on any step?

Here are some related questions to expand on the problem:
1. How do we determine the domain of the inverse function in general?
2. What is the geometric interpretation of an inverse function?
3. How would the answer change if the function were defined on \( (-\infty, 0] \)?
4. Can every function have an inverse? What conditions must hold?
5. What is the significance of the fact that \( f(f^{-1}(x)) = x \)?

**Tip:** When solving for the inverse of a function, it's crucial to consider the domain and range of the original function to determine the valid inverse.

Consider the function f(x) = x^2 - 25 on the interval [0, ∞). Part A: Find the equation for the inverse f^(-1)(x). Part B: Compute f(f^(-1)(x)). Show all work.

Explore the steps to find the inverse of f(x) = x^2 - 25 and compute f(f^(-1)(x)). This solution covers inverse functions, quadratic functions, and function composition. Suitable for high school students studying algebra.

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