For the function g defined by \(\mathrm{g(x) = 5 - |x + 3|}\), what is the maximum value?

1. INFER the optimization strategy

• Given function: \(\mathrm{g(x) = 5 - |x + 3|}\)
• To find maximum of \(\mathrm{g(x)}\), recognize that we're subtracting \(\mathrm{|x + 3|}\) from 5
• Key insight: To maximize \(\mathrm{g(x)}\), we need to minimize \(\mathrm{|x + 3|}\)

2. INFER when absolute value is minimized

• Since \(\mathrm{|x + 3| \geq 0}\) for all real numbers, the minimum possible value is 0
• \(\mathrm{|x + 3| = 0}\) when \(\mathrm{x + 3 = 0}\)
• Solving: \(\mathrm{x = -3}\)

3. SIMPLIFY to find the maximum value

• Substitute \(\mathrm{x = -3}\) into \(\mathrm{g(x)}\):
• \(\mathrm{g(-3) = 5 - |(-3) + 3|}\)
  \(\mathrm{= 5 - |0|}\)
  \(\mathrm{= 5 - 0}\)
  \(\mathrm{= 5}\)

Answer: D (5)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students confuse the x-coordinate that gives the maximum with the actual maximum value.

They correctly identify that \(\mathrm{x = -3}\) makes \(\mathrm{|x + 3| = 0}\), but then mistakenly think the maximum value of the function is -3 instead of evaluating \(\mathrm{g(-3) = 5}\).

This may lead them to select Choice A (-3).

Second Most Common Error Path:

Poor INFER reasoning: Students don't recognize the inverse relationship between \(\mathrm{|x + 3|}\) and \(\mathrm{g(x)}\).

Instead of minimizing \(\mathrm{|x + 3|}\) to maximize \(\mathrm{g(x)}\), they might try to maximize \(\mathrm{|x + 3|}\), leading to confusion about what values to consider or incorrect strategic thinking.

This leads to confusion and guessing among the remaining choices.

The Bottom Line:

The key challenge is recognizing that maximizing a function of the form "constant minus absolute value" requires minimizing the absolute value component. Students who miss this strategic insight often confuse intermediate steps with the final answer.