- Substitute each x-coordinate into the function \(f(x) = 2^x + 1\) to determine the expected y-value.
- Compare the calculated y-value with the given y-coordinate for each ordered pair.
- Identify the pairs that satisfy the equation as being on the graph: \((3, 9), (1, 3), (5, 33)\).
- Identify the pairs that do not satisfy the equation as being off the graph: \((-2, -3), (4, 9)\).
- Final categorization: On the graph: \(\boxed{{(3, 9), (1, 3), (5, 33)}}\); Not on the graph: \(\boxed{{(-2, -3), (4, 9)}}\).
Explanation
- Problem Analysis
To determine if an ordered pair \((x, y)\) lies on the graph of the function \(f(x) = 2^x + 1\), we must verify if substituting the x-coordinate into the function results in the corresponding y-coordinate. That is, the point \((x, y)\) is on the graph if and only if \(y = 2^x + 1\). We will examine each of the five provided ordered pairs: \((3, 9), (-2, -3), (1, 3), (5, 33), \text{ and } (4, 9)\).
- Evaluating the Ordered Pairs
We test each pair by evaluating \(f(x) = 2^x + 1\) at the given x-value and comparing the result with the given y-value:
- For \((3, 9)\): \(f(3) = 2^3 + 1 = 8 + 1 = 9\). Since \(9 = 9\), the point is on the graph.
- For \((-2, -3)\): \(f(-2) = 2^{-2} + 1 = \frac{1}{4} + 1 = 1.25\). Since \(1.25
eq -3\), the point is not on the graph.
- For \((1, 3)\): \(f(1) = 2^1 + 1 = 2 + 1 = 3\). Since \(3 = 3\), the point is on the graph.
- For \((5, 33)\): \(f(5) = 2^5 + 1 = 32 + 1 = 33\). Since \(33 = 33\), the point is on the graph.
- For \((4, 9)\): \(f(4) = 2^4 + 1 = 16 + 1 = 17\). Since \(17
eq 9\), the point is not on the graph.
- Final Classification
Based on our evaluations, we can categorize the pairs as follows:
- On the graph: \((3, 9), (1, 3), (5, 33)\)
- Not on the graph: \((-2, -3), (4, 9)\)
The final categorization is confirmed.
Examples
In data science or computer programming, verifying whether a set of coordinates belongs to a specific model or function is a common task. For instance, when validating if a set of data points follows an exponential growth pattern—such as bacterial growth or population increase modeled by a function like \(f(x) = 2^x + 1\)—we check each data point against the model's equation. If a data point doesn't satisfy the equation, it suggests either noise in the data or that the model needs adjustment to better represent the real-world phenomenon.