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

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

### Step 2: Swap \( x \) and \( y \):
The inverse function is obtained by switching \( x \) and \( y \):
\[ x = (y - 3)^3 \]

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

Now, add 3 to both sides:
\[ y = \sqrt[3]{x} + 3 \]

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

### Step 4: Identify the correct option:
The correct answer is option **D**:
\[ y = \sqrt[3]{x} + 3 \]

Would you like further details on the process or have any questions? Here are 5 related questions to deepen your understanding:

1. What is the definition of an inverse function?
2. How can you verify that two functions are inverses of each other?
3. What is the domain and range of the function \( y = (x - 3)^3 \)?
4. How does the cube root function differ from the square root function in terms of solving for \( y \)?
5. What happens to the graph of a function when you take its inverse?

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

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

Learn how to find the inverse of the function y = (x - 3)^3. This step-by-step guide covers key concepts such as inverse functions, cubic equations, and solving for y. Suitable for high school students in Grades 10-12.

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