- Represent the function as \(y = (x+2)^5\) and solve for \(x\) to find the inverse.
- Swap \(x\) and \(y\) to get \(x = (y+2)^5\), then solve for \(y\) to find \(f^{-1}(x) = \sqrt[5]{x} - 2\).
- Verify the inverse by calculating both \(f(f^{-1}(x))\) and \(f^{-1}(f(x))\), which both simplify to \(x\).
- Conclude that the correct choice is C, with the inverse function \(f^{-1}(x) = \sqrt[5]{x} - 2\) for all \(x\).
- The final answer is \(\boxed{f^{-1}(x) = \sqrt[5]{x} - 2}\).
Explanation
- Analysis of the Problem
The function is given by \(f(x) = (x + 2)^5\). Since this is a power function with an odd exponent (5), it is a one-to-one function across its entire domain, which is all real numbers \((-\infty, \infty)\). Because it is one-to-one, an inverse function \(f^{-1}(x)\) exists. The inverse function will also have a domain and range of all real numbers.
- Finding the Inverse Function
To find the inverse function \(f^{-1}(x)\), we begin by setting the function equal to \(y\):
\[y = (x + 2)^5\]
Next, we swap the roles of \(x\) and \(y\) to reflect the definition of an inverse:
\[x = (y + 2)^5\]
Now, we solve for \(y\) to express it in terms of \(x\). First, we take the fifth root of both sides:
\[\sqrt[5]{x} = \sqrt[5]{(y + 2)^5}\]
\[\sqrt[5]{x} = y + 2\]
Finally, subtracting 2 from both sides gives the inverse function:
\[y = \sqrt[5]{x} - 2\]
Thus, the inverse function is \(f^{-1}(x) = \sqrt[5]{x} - 2\), which is valid for all real \(x\).
- Verification of the Inverse
To verify that \(f^{-1}(x) = \sqrt[5]{x} - 2\) is indeed the inverse of \(f(x) = (x + 2)^5\), we check the compositions \(f(f^{-1}(x))\) and \(f^{-1}(f(x))\).
First, we compute \(f(f^{-1}(x))\):
\[f(f^{-1}(x)) = f(\sqrt[5]{x} - 2) = ((\sqrt[5]{x} - 2) + 2)^5 = (\sqrt[5]{x})^5 = x\]
Next, we compute \(f^{-1}(f(x))\):
\[f^{-1}(f(x)) = f^{-1}((x + 2)^5) = \sqrt[5]{(x + 2)^5} - 2 = (x + 2) - 2 = x\]
Since both compositions equal \(x\), the inverse function is correctly identified as \(f^{-1}(x) = \sqrt[5]{x} - 2\).
- Final Conclusion
Based on our derivation and verification, the correct choice is option C. The inverse function is defined for all real numbers \(x\).
The inverse function is:
\[\boxed{f^{-1}(x) = \sqrt[5]{x} - 2}\]
Examples
In computer programming, inverse functions are essential when you need to "undo" a data transformation. For example, if a program encrypts a piece of data by shifting it through a power function, the receiving end must apply the inverse function to decrypt it and retrieve the original information. Similarly, in fields like engineering or physics, if you have a mathematical model that maps an input value (like force) to an output value (like extension), finding the inverse allows you to calculate the necessary force for a desired extension.