- An inverse function \(g(x) = f^{-1}(x)\) reverses the mapping of \(f(x)\), satisfying \(f(g(x)) = x\).
- Using the provided table, find \(g(2)\) by identifying the input \(x\) such that \(f(x) = 2\), which is \(x = -3\).
- Substitute \(g(2) = -3\) into the expression to get \(f(-3) = 2\).
- The final result is \(\boxed{2}\).
Explanation
- Understanding Inverse Functions
The problem provides a table of values for a function \(f(x)\) and defines \(g(x)\) as its inverse, \(f^{-1}(x)\). An inverse function \(g(x)\) reverses the mapping of \(f(x)\); if \(f(x) = y\), then \(g(y) = x\). The expression we are asked to evaluate is \(f(g(2))\). To solve this, we rely on the fundamental definition of inverse functions: for any \(x\) in the domain of \(g\), the composition \(f(g(x))\) returns the original value \(x\), i.e., \(f(g(x)) = x\).
- Evaluating the Inverse Function at 2
We are given the following values for \(f(x)\):
| \(x\) | \(f(x)\) |
| :--- | :--- |
| \(-6\) | \(1\) |
| \(-3\) | \(2\) |
| \(2\) | \(5\) |
| \(5\) | \(3\) |
| \(8\) | \(0\) |
We need to determine the value of \(f(g(2))\). First, let's find \(g(2)\). Since \(g(x) = f^{-1}(x)\), finding \(g(2)\) is equivalent to finding the value \(x\) such that \(f(x) = 2\). Looking at our table, we see that when \(x = -3\), \(f(-3) = 2\). Therefore, \(g(2) = -3\).
- Calculating the Final Value
Now that we have determined \(g(2) = -3\), we can substitute this back into our original expression:
\(f(g(2)) = f(-3)\).
From the table, we already know that \(f(-3) = 2\). Thus, \(f(g(2)) = 2\). This result is consistent with the property that \(f(f^{-1}(x)) = x\). Since the input to the composition is \(2\), the output must be \(2\).
Examples
In programming and data management, inverse functions are essential when you need to "undo" an operation. For instance, if you have a function that encrypts a piece of data based on a specific key (the forward function), you use the inverse function (decryption) with the same key to retrieve the original data. If you encrypt a number and then immediately decrypt it, you will always arrive back at your starting number, just as \(f(g(x)) = x\). This principle of reversibility is fundamental to secure communication and data retrieval.