Question:The function \(\mathrm{f(y) = 12 - 3|1 - y|}\). What is the value of \(\mathrm{f(4)}\)?

1. TRANSLATE the problem information

• Given: \(\mathrm{f(y) = 12 - 3|1 - y|}\)
• Find: \(\mathrm{f(4)}\)
• What this means: Substitute 4 everywhere you see y in the function

2. SIMPLIFY through substitution

• Replace y with 4: \(\mathrm{f(4) = 12 - 3|1 - 4|}\)
• Work inside the absolute value first: \(\mathrm{1 - 4 = -3}\)
• Now we have: \(\mathrm{f(4) = 12 - 3|-3|}\)

3. SIMPLIFY the absolute value

• Remember: |negative number| = positive number
• So \(\mathrm{|-3| = 3}\)
• Expression becomes: \(\mathrm{f(4) = 12 - 3(3)}\)

4. SIMPLIFY to final answer

• Multiply first: \(\mathrm{3(3) = 9}\)
• Then subtract: \(\mathrm{f(4) = 12 - 9 = 3}\)

Answer: A. 3

Why Students Usually Falter on This Problem

Most Common Error Path:

Conceptual confusion about absolute value: Students think that \(\mathrm{|-3| = -3}\), keeping the negative sign.

Following this incorrect reasoning: \(\mathrm{f(4) = 12 - 3(-3) = 12 - (-9) = 12 + 9 = 21}\)

This leads them to select Choice D (21)

Second Most Common Error:

Weak SIMPLIFY execution: Students rush through the order of operations and subtract 3 from 12 before dealing with the absolute value portion.

They compute \(\mathrm{f(4) = 12 - 3 = 9}\), completely ignoring the |1 - 4| part.

This may lead them to select Choice B (9)

The Bottom Line:

Absolute value problems require careful attention to the definition: absolute value always produces a non-negative result, regardless of what's inside the bars. The key is methodically working through each step without shortcuts.