Consider the following function.
f(x) = x4 + 4,    x ≥ 0
Find an explicit formula for 
f −1.

We are tasked with finding the inverse of the function \( f(x) = x^4 + 4 \) for \( x \geq 0 \).

### Steps to find the inverse function:

1. **Start by replacing \( f(x) \) with \( y \):**
   \[
   y = x^4 + 4
   \]

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

   - Subtract 4 from both sides:
     \[
     y - 4 = x^4
     \]
   - Take the fourth root (since \( x \geq 0 \), we only take the positive root):
     \[
     x = \sqrt[4]{y - 4}
     \]

3. **Express the inverse function:**

   Therefore, the inverse function is:
   \[
   f^{-1}(y) = \sqrt[4]{y - 4}
   \]

Thus, the explicit formula for \( f^{-1}(x) \) is:
\[
f^{-1}(x) = \sqrt[4]{x - 4}
\]

### Final answer:
\[
f^{-1}(x) = \sqrt[4]{x - 4}
\]

Would you like any further details or explanation?

Here are five related questions:
1. What are the domain and range of \( f^{-1}(x) \)?
2. How do we verify if two functions are inverses of each other?
3. What happens to the function if the domain of \( f(x) \) is extended to negative values?
4. Can \( f^{-1}(x) \) be expressed differently in terms of roots?
5. How does this method apply to other polynomial functions?

**Tip:** Always check if the function is one-to-one before finding its inverse.

Consider the following function. f(x) = x^4 + 4, x ≥ 0. Find an explicit formula for f −1.

Learn how to find the inverse of the polynomial function f(x) = x^4 + 4 for x ≥ 0. This detailed solution walks through the steps to derive the inverse function f^{-1}(x) = \sqrt[4]{x - 4}, using key concepts in algebra and inverse functions. Suitable for Grades 10-12.

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