Verified Answer :

  • Identify the input-output pairs from the given table: \((3, 11), (5, 17), (7, 23), (9, 29)\).
  • Substitute the input \(x=3\) into each provided function option to eliminate incorrect candidates.
  • Verify the remaining candidate, \(f(x) = 3x + 2\), by checking it against all other input points in the table.
  • Confirm that \(f(x) = 3x + 2\) satisfies all given pairs and conclude the final answer is \(\boxed{f(x) = 3x + 2}\).

Explanation

  1. Analyzing the Data and Objective

The problem asks us to determine which of the four given linear functions describes the relationship between the inputs \(x\) and the outputs \(f(x)\) provided in the table:

Input (\(x\))3579 \) Output (\(f(x)\))11172329 \)

We are given four candidate functions:
1) \(f(x) = 3x + 1\)
2) \(f(x) = 2x + 5\)
3) \(f(x) = 3x + 2\)
4) \(f(x) = x + 7\)
To find the correct function, we must test each candidate function by substituting the input values \(x \in \{3, 5, 7, 9\}\) and verifying if the calculated output matches the corresponding value in the table. A correct function must satisfy all four input-output pairs simultaneously.

  1. Testing Candidate Functions

We begin testing the candidate functions against the data points.

  • Test \(f(x) = 3x + 1\): For \(x = 3\), \(f(3) = 3(3) + 1 = 10\). Since the table value for \(x=3\) is 11, this function is incorrect.
  • Test \(f(x) = 2x + 5\): For \(x = 3\), \(f(3) = 2(3) + 5 = 11\). This matches the first point. Now test \(x = 5\): \(f(5) = 2(5) + 5 = 15\). Since the table value for \(x=5\) is 17, this function is incorrect.
  • Test \(f(x) = 3x + 2\):
  • For \(x = 3\), \(f(3) = 3(3) + 2 = 9 + 2 = 11\). (Matches)
  • For \(x = 5\), \(f(5) = 3(5) + 2 = 15 + 2 = 17\). (Matches)
  • For \(x = 7\), \(f(7) = 3(7) + 2 = 21 + 2 = 23\). (Matches)
  • For \(x = 9\), \(f(9) = 3(9) + 2 = 27 + 2 = 29\). (Matches)
  • Test \(f(x) = x + 7\): For \(x = 3\), \(f(3) = 3 + 7 = 10\). Since the table value for \(x=3\) is 11, this function is incorrect.
  1. Conclusion

Since the function \(f(x) = 3x + 2\) successfully produces the outputs 11, 17, 23, and 29 for the inputs 3, 5, 7, and 9, respectively, it is the correct function. The other options were eliminated because they did not satisfy all the input-output pairs in the table. Therefore, the function that corresponds to the given table is \(\boxed{f(x) = 3x + 2}\).

Examples

In real-world data analysis, such as predicting a service's hourly cost based on usage, we often have discrete data points and look for a model (like a linear function) that captures the pattern. If a technician charges a base fee plus an hourly rate, the total cost follows a linear relationship \(y = mx + b\), where \(m\) is the hourly rate and \(b\) is the base fee. By comparing given bills for different durations, we can identify the specific charging model using the method demonstrated in this exercise.

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