The function \(\mathrm{f(x) = |50 - 4x|}\). What is the value of \(\mathrm{f(15)}\)?

1. TRANSLATE the problem information

• Given: \(\mathrm{f(x) = |50 - 4x|}\)
• Find: \(\mathrm{f(15)}\)
• This means substitute 15 for every x in the function definition

2. TRANSLATE the substitution

• Replace x with 15: \(\mathrm{f(15) = |50 - 4(15)|}\)
• The absolute value bars tell us to calculate everything inside first

3. SIMPLIFY the expression inside the absolute value

• Calculate \(\mathrm{4(15) = 60}\)
• Now we have: \(\mathrm{f(15) = |50 - 60|}\)
• Calculate \(\mathrm{50 - 60 = -10}\)
• Now we have: \(\mathrm{f(15) = |-10|}\)

4. INFER the final step using absolute value definition

• The absolute value of any number is its distance from zero
• Distance is always positive, so \(\mathrm{|-10| = 10}\)

Answer: B. 10

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students calculate everything correctly up to \(\mathrm{f(15) = |-10|}\), but then think the absolute value of \(\mathrm{-10}\) is still \(\mathrm{-10}\). They don't understand that absolute value always produces a non-negative result.

This leads them to select Choice A (-10).

Second Most Common Error:

Poor TRANSLATE reasoning: Students misread the function or make substitution errors, such as calculating \(\mathrm{|50 - 15|}\) instead of \(\mathrm{|50 - 4(15)|}\), forgetting to multiply by 4.

This calculation gives \(\mathrm{|50 - 15| = |35| = 35}\), leading them to select Choice C (35).

The Bottom Line:

This problem tests whether students truly understand what absolute value means - it's not just "remove the negative sign sometimes," but rather "find the distance from zero, which is always positive."