- Evaluate \(g(x) = (-2)^x - 3\) for \(x \in \{1, 2, 3\}\).
- \(g(1) = -5\), \(g(2) = 1\), \(g(3) = -11\).
- Categorize the given points by comparing them to these values.
- On the graph: \(\{(1, -5), (2, 1), (3, -11)\}\); Not on the graph: \(\{(1, -1), (2, -7), (3, 5)\}\).
Explanation
- Analyzing the Function and Coordinates
To determine which ordered pairs \((x, y)\) lie on the graph of the function \(g(x) = (-2)^x - 3\), we evaluate the function at each given \(x\)-coordinate and compare the result with the corresponding \(y\)-coordinate. A point \((x, y)\) is on the graph if and only if \(y = g(x)\). We will test each unique \(x\)-coordinate provided in the set: \(x=1\), \(x=2\), and \(x=3\).
- Evaluating for x = 1
For \(x = 1\), we calculate the function value:
\[g(1) = (-2)^1 - 3 = -2 - 3 = -5\]
Comparing this result to the given points \((1, -1)\) and \((1, -5)\):
- Since \(g(1) = -5\), the point \((1, -5)\) lies on the graph.
- Since \(g(1)
eq -1\), the point \((1, -1)\) does not lie on the graph.
- Evaluating for x = 2
For \(x = 2\), we calculate the function value:
\[g(2) = (-2)^2 - 3 = 4 - 3 = 1\]
Comparing this result to the given points \((2, 1)\) and \((2, -7)\):
- Since \(g(2) = 1\), the point \((2, 1)\) lies on the graph.
- Since \(g(2)
eq -7\), the point \((2, -7)\) does not lie on the graph.
- Evaluating for x = 3
For \(x = 3\), we calculate the function value:
\[g(3) = (-2)^3 - 3 = -8 - 3 = -11\]
Comparing this result to the given points \((3, 5)\) and \((3, -11)\):
- Since \(g(3) = -11\), the point \((3, -11)\) lies on the graph.
- Since \(g(3)
eq 5\), the point \((3, 5)\) does not lie on the graph.
- Conclusion
Based on our evaluations, we categorize the points as follows:
- On the graph: \((1, -5)\), \((2, 1)\), and \((3, -11)\).
- Not on the graph: \((1, -1)\), \((2, -7)\), and \((3, 5)\).
The final categorization is complete.
Examples
Understanding whether points lie on a function's graph is fundamental in data science and engineering, where we often model real-world phenomena with mathematical equations. For example, if an engineer models the cooling rate of a server component with a function, they must verify if specific temperature readings (points) align with the theoretical model (the graph). By checking if observed data points satisfy the model equation, they can validate the accuracy of their predictions and ensure the system operates within safe parameters.