Let f be the function defined by \(\mathrm{f(x) = |x| - 2}\). Which table gives three values of x and...

1. TRANSLATE the problem information

• Given information:
  • Function: \(\mathrm{f(x) = |x| - 2}\)
  • Need to find f(x) values for x = -3, 0, and 2
• What this tells us: We need to substitute each x-value into the function and calculate the result

2. INFER the approach

• Since we have specific x-values, we substitute each one into \(\mathrm{f(x) = |x| - 2}\)
• Remember: absolute value |x| means the distance from zero, so it's always non-negative
• For negative inputs, \(\mathrm{|-3| = 3}\) (not -3!)

3. SIMPLIFY each calculation

For x = -3:

• \(\mathrm{f(-3) = |-3| - 2}\)
• \(\mathrm{f(-3) = 3 - 2 = 1}\)

For x = 0:

• \(\mathrm{f(0) = |0| - 2}\)
• \(\mathrm{f(0) = 0 - 2 = -2}\)

For x = 2:

• \(\mathrm{f(2) = |2| - 2}\)
• \(\mathrm{f(2) = 2 - 2 = 0}\)

4. TRANSLATE results back to answer format

• The ordered pairs are: \(\mathrm{(-3, 1), (0, -2), (2, 0)}\)
• Compare with the given tables to find the match

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Conceptual confusion about absolute value: Students might think \(\mathrm{|-3| = -3}\) instead of \(\mathrm{|-3| = 3}\)

This leads to calculating \(\mathrm{f(-3) = -3 - 2 = -5}\), which would make them select Choice A (-5, -2, 0).

Second Most Common Error:

Weak INFER skill: Students might confuse the function with similar-looking expressions from the distractors

For example, they might accidentally calculate using \(\mathrm{f(x) = x^2 - 2}\) instead of \(\mathrm{f(x) = |x| - 2}\), leading them to select Choice C (7, -2, 2).

The Bottom Line:

The key insight is remembering that absolute value always produces a non-negative result, regardless of whether the input is positive or negative. This is the crucial step that distinguishes the correct answer from the most tempting distractor.