find the inverse of y= (x^4 -3) for x> 3^1/4

To find the inverse of the function \( y = x^4 - 3 \), we will follow these steps:

1. **Start with the equation:**

   \[
   y = x^4 - 3
   \]

2. **Solve for \( x \) in terms of \( y \):**
   - Add 3 to both sides:
   
     \[
     y + 3 = x^4
     \]
   
   - Take the fourth root of both sides:
   
     \[
     x = \sqrt[4]{y + 3}
     \]

3. **Restrict the domain:**
   Since \( x > \sqrt[4]{3} \), this means we are only considering the positive root of \( x \). Therefore, the inverse function is:

   \[
   x = f^{-1}(y) = \sqrt[4]{y + 3}
   \]

Thus, the inverse function is:

\[
f^{-1}(y) = \sqrt[4]{y + 3}
\]

Let me know if you'd like more details or have questions!

Here are 5 related questions:
1. How do you verify if two functions are inverses?
2. What happens to the domain and range when finding an inverse function?
3. How does the restriction \( x > \sqrt[4]{3} \) affect the inverse function?
4. What if the domain was not restricted? How would the inverse change?
5. How does this process differ when finding the inverse of a cubic function?

**Tip:** Always check the domain and range carefully when finding inverse functions, especially when dealing with even-powered functions like squares or fourth powers.

find the inverse of y = (x^4 - 3) for x > 3^1/4

Learn how to find the inverse of the function y = (x^4 - 3) for x > 3^(1/4). This problem involves key algebraic concepts such as inverse functions and exponents. Suitable for students in Grades 10-12, with a step-by-step explanation of how to solve for x in terms of y and apply domain restrictions.

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