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

1. TRANSLATE the problem information

• Given function: \(\mathrm{g(x) = 5 - |x + 3|}\)
• Need to find: Maximum value of \(\mathrm{g(x)}\)

2. INFER the optimization strategy

• The function has the form: constant - absolute value
• To maximize this, we need to minimize what we're subtracting
• Key insight: We need to make \(\mathrm{|x + 3|}\) as small as possible

3. APPLY CONSTRAINTS to find minimum absolute value

• By definition, absolute values are non-negative: \(\mathrm{|x + 3| \geq 0}\)
• The smallest possible value of \(\mathrm{|x + 3|}\) is 0
• This occurs when \(\mathrm{x + 3 = 0}\), so \(\mathrm{x = -3}\)

4. Calculate the maximum value

• When \(\mathrm{x = -3}\): \(\mathrm{g(-3) = 5 - |(-3) + 3|}\)
  \(\mathrm{= 5 - 0}\)
  \(\mathrm{= 5}\)
• Therefore, the maximum value of \(\mathrm{g(x)}\) is 5

Answer: D (5)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the optimization strategy for functions of the form "constant - absolute value." Instead, they might try calculus approaches or randomly test values from the answer choices. This leads to confusion and guessing among the available options.

Second Most Common Error:

Poor TRANSLATE reasoning: Students might confuse "maximum value of the function" with "x-value where maximum occurs." They correctly find that the maximum occurs at \(\mathrm{x = -3}\), but then select Choice A (-3) instead of recognizing that -3 is the input, not the output value we're looking for.

The Bottom Line:

This problem tests whether students understand how absolute value constraints affect function optimization. The key breakthrough is recognizing that minimizing the absolute value term maximizes the overall function—a strategic insight that many students miss.