The table below shows values of a function f. Which of the following defines f?x\(\mathrm{f(x)}\)051225\(\mathrm{f(x) = x + 5}\)\(\mathrm{f(x) =...

1. TRANSLATE the problem information

• Given information:
  • Table with three data points: \(\mathrm{(0,5)}\), \(\mathrm{(1,2)}\), \(\mathrm{(2,5)}\)
  • Four possible function definitions to test
• What we need: Find which function produces all three outputs correctly

2. INFER the approach

• Key insight: A function must work for ALL given points, not just some
• Strategy: Test each function systematically against every data point
• Warning: Don't stop testing after finding functions that work for one or two points

3. SIMPLIFY by testing each function

Testing Option A: \(\mathrm{f(x) = x + 5}\)

• \(\mathrm{f(0) = 0 + 5 = 5}\) ✓ (matches table)
• \(\mathrm{f(1) = 1 + 5 = 6 ≠ 2}\) ✗ (doesn't match table)
• Option A is eliminated - no need to test \(\mathrm{f(2)}\)

Testing Option B: \(\mathrm{f(x) = 5 - 3x}\)

• \(\mathrm{f(0) = 5 - 3(0) = 5}\) ✓
• \(\mathrm{f(1) = 5 - 3(1) = 2}\) ✓
• \(\mathrm{f(2) = 5 - 3(2) = -1 ≠ 5}\) ✗
• Option B is eliminated

Testing Option C: \(\mathrm{f(x) = x² - 4x + 5}\)

• \(\mathrm{f(0) = (0)² - 4(0) + 5 = 5}\) ✓
• \(\mathrm{f(1) = (1)² - 4(1) + 5 = 1 - 4 + 5 = 2}\) ✓
• \(\mathrm{f(2) = (2)² - 4(2) + 5 = 4 - 8 + 5 = 1 ≠ 5}\) ✗
• Option C is eliminated

4. CONSIDER ALL CASES by testing the final option

Testing Option D: \(\mathrm{f(x) = 3x² - 6x + 5}\)

• \(\mathrm{f(0) = 3(0)² - 6(0) + 5 = 5}\) ✓
• \(\mathrm{f(1) = 3(1)² - 6(1) + 5 = 3 - 6 + 5 = 2}\) ✓
• \(\mathrm{f(2) = 3(2)² - 6(2) + 5 = 12 - 12 + 5 = 5}\) ✓

All three points match! Option D is our answer.

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak CONSIDER ALL CASES skill: Students test functions against only the first data point or first two points, then select an answer without complete verification.

For example, they might see that options A, B, C, and D all work for \(\mathrm{f(0) = 5}\), then notice B, C, and D work for \(\mathrm{f(1) = 2}\), and incorrectly conclude that any of these remaining options could be correct. They might guess or pick the "simplest looking" option like B.

This may lead them to select Choice B (\(\mathrm{f(x) = 5 - 3x}\)) without testing \(\mathrm{f(2)}\).

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors when evaluating the polynomial expressions, especially with the quadratic functions involving multiple operations.

Common mistakes include sign errors in subtraction, forgetting to square the x-value first, or mixing up order of operations. These calculation errors can make correct functions appear wrong or wrong functions appear correct.

This leads to confusion and guessing among the remaining options.

The Bottom Line:

This problem tests patience and systematic verification. Success requires testing every function against every data point - there are no shortcuts that guarantee the right answer.