The function p is defined by \(\mathrm{p(x) = x + k}\), where k is a constant. If \(\mathrm{p(p(0)) = 60}\),...

1. TRANSLATE the problem information

• Given information:
  • Function \(\mathrm{p(x) = x + k}\) (where \(\mathrm{k}\) is unknown constant)
  • \(\mathrm{p(p(0)) = 60}\)
• What this tells us: We need to evaluate a composite function and use the result to find \(\mathrm{k}\)

2. INFER the solution approach

• Key insight: Composite functions like \(\mathrm{p(p(0))}\) must be evaluated from inside out
• Strategy: First find \(\mathrm{p(0)}\), then use that result as input for the outer \(\mathrm{p}\) function

3. SIMPLIFY the inner function first

• Calculate \(\mathrm{p(0)}\): \(\mathrm{p(0) = 0 + k = k}\)
• So \(\mathrm{p(p(0)) = p(k)}\)

4. SIMPLIFY the outer function

• Calculate \(\mathrm{p(k)}\): \(\mathrm{p(k) = k + k = 2k}\)
• Therefore \(\mathrm{p(p(0)) = 2k}\)

5. SIMPLIFY to solve for k

• Set up equation: \(\mathrm{2k = 60}\)
• Solve: \(\mathrm{k = 30}\)

Answer: C (30)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skills: Students misinterpret \(\mathrm{p(p(0))}\) and try to substitute directly without working step-by-step

Instead of evaluating \(\mathrm{p(0)}\) first, they might write something like \(\mathrm{p(p(0)) = (0 + k) + k = 2k}\), which happens to give the right setup but shows they're not truly understanding the function composition process. More problematically, they might try \(\mathrm{p(p(0)) = p(0 + k)}\) and get confused about what to do next.

This leads to confusion and potentially guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{2k = 60}\) but make arithmetic errors

They might solve \(\mathrm{2k = 60}\) incorrectly, perhaps getting \(\mathrm{k = 120}\) (multiplying instead of dividing) or \(\mathrm{k = 58}\) (subtracting 2 instead of dividing by 2). Since none of the answer choices match these errors exactly, this causes them to get stuck and guess.

The Bottom Line:

Function composition requires systematic inside-out evaluation. Students who try to take shortcuts or work outside-in will struggle with the logical sequence needed to reach the correct answer.