What is the inverse function of ⁵√2x+4

To find the inverse of the function \( f(x) = \sqrt[5]{2x + 4} \), we follow these steps:

### Step 1: Replace \( f(x) \) with \( y \)
\[
y = \sqrt[5]{2x + 4}
\]

### Step 2: Swap \( x \) and \( y \)
To find the inverse, swap \( x \) and \( y \):
\[
x = \sqrt[5]{2y + 4}
\]

### Step 3: Solve for \( y \)
Now, solve for \( y \):
1. Raise both sides to the power of 5 to eliminate the fifth root:
   \[
   x^5 = 2y + 4
   \]
2. Subtract 4 from both sides:
   \[
   x^5 - 4 = 2y
   \]
3. Divide by 2:
   \[
   y = \frac{x^5 - 4}{2}
   \]

### Final Inverse Function
The inverse function is:
\[
f^{-1}(x) = \frac{x^5 - 4}{2}
\]

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Do you need more details or have any questions?

Here are 5 related questions:
1. How do you verify if two functions are inverses of each other?
2. What are the domain and range of the inverse function?
3. Can all functions have an inverse? Why or why not?
4. What happens if the function is not one-to-one? 
5. How does the process change for inverse functions of higher-degree polynomials?

**Tip**: To confirm that two functions are inverses, compose them \( f(f^{-1}(x)) \) and \( f^{-1}(f(x)) \). Both should equal \( x \).

What is the inverse function of ⁵√2x+4?

Find the inverse function of ⁵√(2x + 4) with a detailed step-by-step solution. Learn how to solve radical equations, apply exponentiation, and determine the inverse of functions in this algebraic problem. Suitable for students in Grades 10-12.

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