how do you solve f^{-1} for \sqrt[3]{x}+4

To solve for the inverse \( f^{-1}(x) \) of the function \( f(x) = \sqrt[3]{x} + 4 \), follow these steps:

### 1. **Write the function in terms of \( y \):**
   Set \( y = f(x) \). So,
   \[
   y = \sqrt[3]{x} + 4
   \]

### 2. **Solve for \( x \):**
   To find the inverse, you need to solve for \( x \) in terms of \( y \).
   - Subtract 4 from both sides:
     \[
     y - 4 = \sqrt[3]{x}
     \]
   - Cube both sides to eliminate the cube root:
     \[
     (y - 4)^3 = x
     \]

### 3. **Express \( x \) in terms of \( y \):**
   Now you have:
   \[
   x = (y - 4)^3
   \]

### 4. **Rewrite as the inverse function \( f^{-1}(x) \):**
   Replace \( y \) with \( x \) to express the inverse function:
   \[
   f^{-1}(x) = (x - 4)^3
   \]

So, the inverse of \( f(x) = \sqrt[3]{x} + 4 \) is:
\[
f^{-1}(x) = (x - 4)^3
\]

Would you like more details on this process or have any questions?

### Related Questions:
1. How do you find the inverse of a quadratic function?
2. What are the properties of inverse functions?
3. How do you verify that two functions are inverses of each other?
4. How do cube roots affect the domain and range of functions?
5. Can you explain how to solve for inverses of trigonometric functions?

### Tip:
When finding an inverse, always check that the original function is one-to-one (passes the horizontal line test) to ensure the inverse exists.

Learn how to find the inverse of the function f(x) = \sqrt[3]{x} + 4 with a step-by-step explanation. This solution covers key algebraic concepts, including cube roots and the inverse function theorem. Suitable for high school students in Grades 10-12.

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