- Systematically inspect the provided table row by row.
- Compare the output values of \(f(x)\) and \(g(x)\) for each given value of \(x\).
- Identify the input \(x\) where the outputs of the functions are equal.
- Conclude that the solution is \(x = 2\), which corresponds to option c. Final answer: \(\boxed{2}\).
Explanation
- Analyzing the Provided Data
To find a solution to the equation \(f(x) = g(x)\) using the provided table, we evaluate the functions for each given value of \(x\) and determine where the outputs are identical. The table lists the following values:
| \(x\) | \(f(x)\) | \(g(x)\) |
| :--- | :--- | :--- |
| \(0\) | \(-1\) | \(5\) |
| \(1\) | \(1\) | \(4\) |
| \(2\) | \(3\) | \(3\) |
| \(3\) | \(5\) | \(2\) |
| \(4\) | \(7\) | \(3\) |
- Evaluating the Functions at Each Point
We proceed row by row to compare the outputs of \(f(x)\) and \(g(x)\):
- For \(x = 0\): \(f(0) = -1\) and \(g(0) = 5\). Since \(-1
eq 5\), this is not a solution.
- For \(x = 1\): \(f(1) = 1\) and \(g(1) = 4\). Since \(1
eq 4\), this is not a solution.
- For \(x = 2\): \(f(2) = 3\) and \(g(2) = 3\). Since \(3 = 3\), \(x = 2\) is a solution to the equation \(f(x) = g(x)\).
- For \(x = 3\): \(f(3) = 5\) and \(g(3) = 2\). Since \(5
eq 2\), this is not a solution.
- For \(x = 4\): \(f(4) = 7\) and \(g(4) = 3\). Since \(7
eq 3\), this is not a solution.
- Conclusion
Based on our comparison, the only value of \(x\) from the given set that satisfies \(f(x) = g(x)\) is \(x = 2\). This matches option c in the provided choices. Therefore, the solution is \(\boxed{2}\).
Examples
In real-world applications, such as business analysis or engineering, tables of values are often used to compare the performance of two different processes. If \(f(x)\) represents the cost of producing \(x\) units using Method A and \(g(x)\) represents the cost using Method B, finding the intersection point \(f(x) = g(x)\) tells the analyst the specific number of units where the production cost is identical for both methods. This is a fundamental step in making data-driven decisions about resource allocation and process optimization.