Math Problem Statement

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

Solution

To find the inverse of the function y=(x3)3y = (x - 3)^3, we will follow the steps for determining the inverse of a function:

Step 1: Start with the given function:

y=(x3)3y = (x - 3)^3

Step 2: Swap xx and yy:

The inverse function is obtained by switching xx and yy: x=(y3)3x = (y - 3)^3

Step 3: Solve for yy:

To solve for yy, take the cube root of both sides: x3=y3\sqrt[3]{x} = y - 3

Now, add 3 to both sides: y=x3+3y = \sqrt[3]{x} + 3

Thus, the inverse function is: y=x3+3y = \sqrt[3]{x} + 3

Step 4: Identify the correct option:

The correct answer is option D: y=x3+3y = \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=(x3)3y = (x - 3)^3?
  4. How does the cube root function differ from the square root function in terms of solving for yy?
  5. What happens to the graph of a function when you take its inverse?

Tip: Always remember to swap xx and yy when finding the inverse of a function, and then solve for yy.

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Math Problem Analysis

Mathematical Concepts

Inverse Functions
Cubic Functions
Algebra

Formulas

y = (x - 3)^3
y = cube root of x + 3

Theorems

Inverse Function Theorem

Suitable Grade Level

Grades 10-12