- Understand the property of inverse functions: \(f(g(x)) = x\).
- Recognize that the domain of \(g(x)\) is the range of \(f(x)\), and since \(2\) is in the range of \(f(x)\), the expression \(f(g(2))\) is defined.
- Identify from the table that \(f(-3) = 2\), which implies \(g(2) = -3\).
- Calculate \(f(g(2)) = f(-3) = 2\).
- Final Answer: \(\boxed{2}\)
Explanation
- Analysis of the Problem
The problem asks for the value of the composite function \(f(g(2))\), where \(f(x)\) is a function given by a set of points in a table, and \(g(x)\) is the inverse function of \(f(x)\), denoted as \(g(x) = f^{-1}(x)\). The provided table lists the following pairs \((x, f(x))\): \((-6, 1), (-3, 2), (2, 5), (5, 3), (8, 0)\).
- Applying the Property of Inverse Functions
By definition, if \(g(x)\) is the inverse of \(f(x)\), then \(f(g(x)) = x\) for every \(x\) in the domain of \(g\). The domain of the inverse function \(g(x)\) is the range of the original function \(f(x)\). From the given table, the range of \(f(x)\) is \(\{1, 2, 5, 3, 0\}\). Since \(2\) is in the range of \(f(x)\), the value \(g(2)\) is well-defined. According to the property of inverse functions, the expression \(f(g(2))\) should simplify directly to \(2\).
- Verification using the Function Table
To verify, we can determine the value of \(g(2)\) first. By definition, \(g(2)\) is the input value \(x\) such that \(f(x) = 2\). Looking at the table, we see that when \(x = -3\), \(f(-3) = 2\). Thus, \(g(2) = -3\). Now, substitute this result back into the expression: \(f(g(2)) = f(-3)\). Referring to the table once more, \(f(-3) = 2\). Therefore, \(f(g(2)) = 2\).
Examples
Inverse functions are frequently used in fields like data science and cryptography. For example, if a function represents an encryption process (mapping plaintext to ciphertext), its inverse function represents the decryption process (mapping ciphertext back to the original plaintext). By applying the original function to the result of its inverse, you return to your starting point, much like how \(f(g(2))\) brought us back to the original input of \(2\). This property ensures data integrity and accurate retrieval of original information.