- Review the table of values for \(f(x)\) and \(g(x)\) for all given inputs.
- Compare the outputs of \(f(x)\) and \(g(x)\) at each \(x\) to see where they are identical.
- Notice that at \(x = 2\), both functions have a value of \(3\).
- Conclude that the solution is \(\boxed{2}\).
Explanation
- Analyzing the problem data
To find the solution to the equation \(f(x) = g(x)\), we examine the provided table of values for both functions at each given point \(x \in \{0, 1, 2, 3, 4\}\). We need to identify the specific value of \(x\) where the outputs of \(f(x)\) and \(g(x)\) are equal.
- Checking the condition \(f(x) = g(x)\)
We evaluate the functions for each value of \(x\):
- For \(x=0\): \(f(0)=-1\) and \(g(0)=5\). Since \(-1
eq 5\), \(x=0\) is not a solution.
- For \(x=1\): \(f(1)=1\) and \(g(1)=4\). Since \(1
eq 4\), \(x=1\) is not a solution.
- For \(x=2\): \(f(2)=3\) and \(g(2)=3\). Since \(3 = 3\), \(x=2\) is a solution.
- For \(x=3\): \(f(3)=5\) and \(g(3)=2\). Since \(5
eq 2\), \(x=3\) is not a solution.
- For \(x=4\): \(f(4)=7\) and \(g(4)=3\). Since \(7
eq 3\), \(x=4\) is not a solution.
- Conclusion
Based on the evaluation of the table, the condition \(f(x) = g(x)\) is satisfied only when \(x = 2\). Therefore, the correct option is c. The final answer is \(\boxed{2}\).
Examples
In many fields, such as economics or physics, we often compare two different trends or models represented by tables of data. Finding where these models intersect—like calculating the "break-even point" where costs equal revenue, or finding when two moving objects meet—is a fundamental skill. By comparing data points in a table, we can identify these moments of intersection easily without needing complex mathematical formulas.