a cubic root function has a domain of x greater than or equal to 9 and a range of y greater than or equal to -3, whats the domain of its inverse

Given that the cubic root function has a domain of \( x \geq 9 \) and a range of \( y \geq -3 \), we can find the domain of its inverse by understanding how the domain and range of the original function relate to the inverse function.

For any function, the domain of the inverse function is the range of the original function, and the range of the inverse function is the domain of the original function.

1. **Domain of the cubic root function**: \( x \geq 9 \).
2. **Range of the cubic root function**: \( y \geq -3 \).

The inverse function will swap these, so:
- **Domain of the inverse function**: \( y \geq -3 \) (the range of the original function).
- **Range of the inverse function**: \( x \geq 9 \) (the domain of the original function).

Thus, the **domain of the inverse function** is \( y \geq -3 \).

Would you like more details or have any questions?

Here are some related questions:
1. How do you find the inverse of a cubic function algebraically?
2. What does the graph of a cubic function and its inverse look like?
3. How do domain and range change when finding the inverse of a general function?
4. What happens to the domain and range if the original cubic root function is modified?
5. How do you determine if an inverse function exists for a given function?

**Tip**: When finding the inverse of a function, always remember that the domain of the original function becomes the range of the inverse and vice versa!

Understand the domain of the inverse function for a cubic root function with a domain of x ≥ 9 and a range of y ≥ -3. Learn how the domain and range of a function relate to its inverse, essential for high school mathematics.

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