- We evaluate each of the four provided candidate functions by substituting the \(x\) values from the table.
- The function \(g(x) = \sqrt{x} + 1\) produces predicted values that match the table values (2.73, 3.45, 4, 4.46, 4.87) with high precision.
- Other functions like \(g(x) = 1.58 \sqrt{x}\) and \(g(x) = \frac{4}{3} \sqrt{x}\) do not match the data points.
- The best-fit model is \(\boxed{g(x) = \sqrt{x} + 1}\).
Explanation
- Analyzing the Data and Candidates
To identify the correct function, we compare the given data points \((x, y)\) in the table with the values predicted by each of the four candidate functions: \(g(x) = \sqrt{x} + 1\), \(g(x) = 1.58 \sqrt{x}\), \(g(x) = \frac{4}{3} \sqrt{x}\), and \(g(x) = \sqrt{x} - 1\). The data points provided are: \((3, 2.73)\), \((6, 3.45)\), \((9, 4)\), \((12, 4.46)\), and \((15, 4.87)\).
- Evaluating and Comparing Functions
We evaluate each candidate function at the given values of \(x\) and compare the results with the corresponding \(y\) values:
- For \(g(x) = \sqrt{x} + 1\):
- At \(x=3\), \(g(3) = \sqrt{3} + 1 \approx 1.732 + 1 = 2.732\).
- At \(x=6\), \(g(6) = \sqrt{6} + 1 \approx 2.449 + 1 = 3.449\).
- At \(x=9\), \(g(9) = \sqrt{9} + 1 = 3 + 1 = 4\).
- At \(x=12\), \(g(12) = \sqrt{12} + 1 \approx 3.464 + 1 = 4.464\).
- At \(x=15\), \(g(15) = \sqrt{15} + 1 \approx 3.873 + 1 = 4.873\).
These values match the table entries very closely (e.g., \(2.73\), \(3.45\), \(4\), \(4.46\), \(4.87\)). Testing the other functions shows significant deviations, such as for \(g(x) = 1.58 \sqrt{x}\) or \(g(x) = \frac{4}{3} \sqrt{x}\), which do not match the data points across all values.
- Conclusion
Based on our analysis, the function \(g(x) = \sqrt{x} + 1\) is the only one that consistently produces the values provided in the data table for all given inputs. Therefore, it is the best-fitting square root model. The final answer is \(\boxed{g(x) = \sqrt{x} + 1}\).
Examples
Modeling data with simple functions like square roots is a foundational skill in statistics and engineering. For example, if you are analyzing the growth of a plant over several weeks (where \(x\) is time and \(y\) is the height), you might find that your measurements align with a specific growth curve. By identifying the mathematical function that best represents these points, you can make predictions about the plant's future height or understand the underlying growth rate. Similar methods are used in physics to model velocity versus time, or in business to predict revenue trends based on limited samples.