- Inspect the provided table to compare \(f(x) = 2^{-x}\) and \(g(x) = -2x\) for every given \(x\).
- Identify rows where the values of \(f(x)\) and \(g(x)\) are identical.
- Determine the \(x\)-values for these matches as the solutions to the equation \(f(x) = g(x)\).
- Conclude that the solutions are \(\boxed{-2, -1}\).
Explanation
- Analysis of the Given Data
To determine the solutions to the equation \(f(x) = g(x)\), we must find the values of \(x\) for which the outputs of both functions are identical. We are given two functions, \(f(x) = 2^{-x}\) and \(g(x) = -2x\), and a table of their values over the interval \(x \in \{-3, -2, -1, 0, 1, 2, 3\}\). By inspecting the table, we can directly compare the outputs for each \(x\).
- Evaluating the Equation at Each Point
We examine the table row by row to find where \(f(x)\) equals \(g(x)\):
- For \(x = -3\): \(f(-3) = 8\) and \(g(-3) = 6\). Since \(8
eq 6\), this is not a solution.
- For \(x = -2\): \(f(-2) = 4\) and \(g(-2) = 4\). Since \(4 = 4\), \(x = -2\) is a solution.
- For \(x = -1\): \(f(-1) = 2\) and \(g(-1) = 2\). Since \(2 = 2\), \(x = -1\) is a solution.
- For \(x = 0\): \(f(0) = 1\) and \(g(0) = 0\). Since \(1
eq 0\), this is not a solution.
- For \(x = 1\): \(f(1) = 0.5\) and \(g(1) = -2\). Since \(0.5
eq -2\), this is not a solution.
- For \(x = 2\): \(f(2) = 0.25\) and \(g(2) = -4\). Since \(0.25
eq -4\), this is not a solution.
- For \(x = 3\): \(f(3) = 0.125\) and \(g(3) = -6\). Since \(0.125
eq -6\), this is not a solution.
- Conclusion
Based on our comparison of the function values in the table, the equation \(f(x) = g(x)\) is satisfied when \(x = -2\) and \(x = -1\). These points represent the intersections of the graphs of \(f(x)\) and \(g(x)\). Thus, the solutions to the equation are \(x = -2\) and \(x = -1\). The final answer is \(\boxed{-2, -1}\).
Examples
Graphing systems of equations is essential in fields like engineering and economics. For example, if you are modeling the profit of a company where one function represents revenue and another represents costs, the point where the two graphs intersect (where revenue equals cost) identifies the "break-even point." Finding these intersections allows decision-makers to determine the production levels needed to achieve profitability or to understand the threshold where one model outperforms another.