- The problem asks for the evaluation of the composition of a function \(f\) and its inverse \(g\) at a specific point: \(f(g(2))\).
- By the fundamental property of inverse functions, \(f(g(y)) = y\) for any \(y\) in the range of \(f\).
- Using the table, we confirm that \(f(-3) = 2\), which implies \(g(2) = -3\).
- Substituting \(g(2) = -3\) back into \(f\) gives \(f(-3) = 2\), so the final result is \(\boxed{2}\).
Explanation
- Analyzing the Function and Inverse Definition
The function \(f(x)\) is given by a set of points: \((-6, 1), (-3, 2), (2, 5), (5, 3),\) and \((8, 0)\). This means, for example, that \(f(-3) = 2\). The function \(g(x)\) is defined as the inverse of \(f(x)\), which means that if \(f(a) = b\), then \(g(b) = a\). We are asked to find the value of the composition \(f(g(2))\).
- Applying the Property of Inverse Functions
By the definition of an inverse function, for any value \(y\) in the domain of \(g\) (which is the range of \(f\)), we have \(f(g(y)) = y\). This is because the inverse function \(g\) undoes the operation performed by \(f\). Specifically, if we start with \(y\), apply \(g\) to get \(x = g(y)\), and then apply \(f\) to \(x\), we return to the original value \(y\). Thus, \(f(g(y)) = y\) holds for all \(y\) in the range of \(f\).
- Evaluating the Expression
We are asked to evaluate \(f(g(2))\). Using the fundamental property established in the previous step, \(f(g(y)) = y\), we simply set \(y = 2\). Therefore, \(f(g(2)) = 2\). To verify this using the provided table:
- Find \(g(2)\): We look for an \(x\) such that \(f(x) = 2\). From the table, \(f(-3) = 2\), so \(g(2) = -3\).
- Evaluate \(f(g(2))\): Substitute \(g(2) = -3\) into the expression: \(f(-3)\).
- From the table, \(f(-3) = 2\).
Thus, \(f(g(2)) = 2\), which matches our expected result.
- Final Conclusion
Based on the evaluation, the value of \(f(g(2))\) is 2. This matches the third option provided in the problem. The correct answer is \(\boxed{2}\).
Examples
In real-life scenarios, inverse functions are frequently used in encryption and decryption. If an encryption function \(f\) transforms a piece of data (the original message) into an encrypted code, the decryption function \(g\) is the inverse of \(f\). When you receive the encrypted code \(g(y)\) and process it through the decryption mechanism \(f\), you effectively perform the operation \(f(g(y))\), which should return you to the original piece of data \(y\). This ensures that information can be secured and later accessed by the intended recipient.