- Examine the provided tables for \(f(x)\) and \(g(x)\).
- Test each candidate input value \(x \in \{-3, -1, 0, 1\}\) to find where \(f(x) = g(x)\).
- Observe that for \(x = 1\), both functions output \(3\), confirming that the equality holds.
- Final answer: \(\boxed{1}\).
Explanation
- Analysis of the problem
To determine which input value \(x\) produces the same output for both functions \(f(x)\) and \(g(x)\), we must compare the function values given in the provided tables for each candidate input value: \(x \in \{-3, -1, 0, 1\}\). We examine each case individually.
- Comparison of function values
We evaluate \(f(x)\) and \(g(x)\) for each candidate input from the options provided:
- For \(x = -3\): From the table, \(f(-3) = -5\) and \(g(-3) = -13\). Since \(-5
eq -13\), \(x = -3\) is not the solution.
- For \(x = -1\): From the table, \(f(-1) = -1\) and \(g(-1) = -5\). Since \(-1
eq -5\), \(x = -1\) is not the solution.
- For \(x = 0\): From the table, \(f(0) = 1\) and \(g(0) = -1\). Since \(1
eq -1\), \(x = 0\) is not the solution.
- For \(x = 1\): From the table, \(f(1) = 3\) and \(g(1) = 3\). Since \(3 = 3\), \(x = 1\) is the correct solution.
- Conclusion
By comparing the outputs for all given candidates, we found that only at \(x = 1\) do the two functions have the same output value of \(3\). Thus, the correct input value is \(1\). The final answer is \(\boxed{1}\).
Examples
In real-life scenarios, comparing two functions can help determine when two different processes or systems reach the same state. For example, if you are comparing two subscription plans where \(f(x)\) represents the cost of plan A after \(x\) months and \(g(x)\) represents the cost of plan B after \(x\) months, finding the point where \(f(x) = g(x)\) tells you at which month both plans become equal in price. This allows you to make informed decisions about which plan is more economical based on how long you intend to use the service.