Which of the following is the inverse of y=(x−3)3y=(x−3)3?

A.y=x−3−−−−−√3y=x−33

B.y=x+3−−−−−√3y=x+33

C.y=x−−√3−3y=x3−3

D.y=x−−√3+3y=x3+3

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

1. Start with the given function:  
   \[ y = (x - 3)^3 \]

2. Swap \( x \) and \( y \) to express the inverse function:
   \[ x = (y - 3)^3 \]

3. Solve for \( y \). First, take the cube root of both sides:
   \[ \sqrt[3]{x} = y - 3 \]

4. Add 3 to both sides:
   \[ y = \sqrt[3]{x} + 3 \]

Thus, the inverse function is:
\[
y = \sqrt[3]{x} + 3
\]

Looking at the options provided, the correct answer is:

**D. \( y = \sqrt[3]{x} + 3 \)**

Would you like further details or clarifications?

### 5 Related Questions:
1. How can you verify that two functions are inverses of each other?
2. What is the graphical relationship between a function and its inverse?
3. How do you find the inverse of a more complex function, such as \( y = (x + 5)^4 \)?
4. What are the restrictions for a function to have an inverse?
5. How does the derivative of a function relate to the derivative of its inverse?

**Tip**: When finding the inverse of a function, always remember to swap \( x \) and \( y \), then solve for \( y \).

Which of the following is the inverse of y = (x − 3)^3?

Discover how to find the inverse of the function y = (x - 3)^3 with detailed steps. This solution explores key algebraic concepts such as cube roots and inverse functions, suitable for students in Grades 9-11.

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