Which of the following is the inverse of y=(x−3)3?A.y=x−3−−−−−√3B.y=x+3−−−−−√3C.y=x−−√3−3D.y=x−−√3+3

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

1. **Rewrite the function** in terms of \( y \):
   \[
   y = (x - 3)^3
   \]

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

3. **Solve for \( y \)**:
   - Take the cube root of both sides:
     \[
     \sqrt[3]{x} = y - 3
     \]
   - Add 3 to both sides:
     \[
     y = \sqrt[3]{x} + 3
     \]

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

The correct answer is:  
**D.** \( y = \sqrt[3]{x} + 3 \)

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Would you like further details on these steps or have any other questions?

Here are 5 related questions to explore further:

1. How do you verify that a function and its inverse are correct?
2. What is the graphical relationship between a function and its inverse?
3. How does the domain and range of a function change after finding the inverse?
4. Can all functions have inverses? Why or why not?
5. How do you find the inverse of more complex functions like \( y = (x^2 - 4) \)?

**Tip:** A function's inverse, if it exists, "undoes" the action of the original function—this is like reversing an operation step-by-step.

Which of the following is the inverse of y=(x−3)^3? A. y=∛(x−3) B. y=∛(x+3) C. y=∛(x)−3 D. y=∛(x)+3

Discover how to find the inverse of the function y = (x−3)^3 with step-by-step instructions. This solution involves key algebraic concepts such as inverse functions and cubic functions, making it suitable for students in Grades 9-12.

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