- Set \(y = (x - 3)^3 + 2\).
- Swap \(x\) and \(y\) to get \(x = (y - 3)^3 + 2\).
- Solve for \(y\) by subtracting 2, taking the cube root, and adding 3.
- The inverse function is \(\boxed{f^{-1}(x) = \sqrt[3]{x - 2} + 3}\).
Explanation
- Analysis of the Problem
The objective is to find the inverse function of \(f(x) = (x - 3)^3 + 2\). To find the inverse of a function \(y = f(x)\), we switch the roles of \(x\) and \(y\) and then solve for \(y\). The resulting expression will be the inverse function \(f^{-1}(x)\). Given that the original function is a cubic function, it is one-to-one over the real numbers, ensuring that its inverse is well-defined.
- Solving for the Inverse Function
We begin with the equation for the given function:
\[y = (x - 3)^3 + 2\]
To find the inverse, we swap \(x\) and \(y\):
\[x = (y - 3)^3 + 2\]
Now, we solve this equation for \(y\). First, subtract 2 from both sides:
\[x - 2 = (y - 3)^3\]
Next, take the cube root of both sides to isolate the expression with \(y\):
\[\sqrt[3]{x - 2} = y - 3\]
Finally, add 3 to both sides to solve for \(y\):
\[y = \sqrt[3]{x - 2} + 3\]
Thus, the inverse function is \(f^{-1}(x) = \sqrt[3]{x - 2} + 3\).
- Conclusion
Comparing our derived inverse function \(f^{-1}(x) = \sqrt[3]{x - 2} + 3\) to the provided options:
- \(f^{-1}(x) = \sqrt[3]{x - 2} + 3\)
- \(f^{-1}(x) = \sqrt[3]{x + 3} - 2\)
- \(f^{-1}(x) = \sqrt[3]{x + 2} - 3\)
- \(f^{-1}(x) = \sqrt[3]{x - 3} + 7\)
Our result matches the first option. Therefore, the correct inverse function is \(f^{-1}(x) = \sqrt[3]{x - 2} + 3\). \(\boxed{f^{-1}(x) = \sqrt[3]{x - 2} + 3}\).
Examples
In many real-world applications, such as data encryption or computer programming, we often need to reverse a transformation process. For example, if a computer program transforms an input \(x\) by subtracting 3, cubing the result, and adding 2, a reverse program needs to "undo" these steps to retrieve the original input. This is exactly what an inverse function does. By calculating the inverse, we can derive the original value from the transformed result, which is fundamental to error correction and decoding operations.