Question:The functions p and q are defined for all real numbers.\(\mathrm{p(t)}\) = {(t + 7)/2, if t lt 03t -...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{p(t)}\) is a piecewise function with two pieces based on whether \(\mathrm{t \lt 0}\) or \(\mathrm{t ≥ 0}\)
  • \(\mathrm{q(s) = |5 - 2s| + s^2}\)
  • We need to find \(\mathrm{k = p(-3)}\), then find \(\mathrm{q(k)}\)
• What this tells us: This is a function composition problem requiring two separate evaluations

2. INFER the approach

• Since we need \(\mathrm{q(k)}\) and \(\mathrm{k = p(-3)}\), we must:
  1. First evaluate \(\mathrm{p(-3)}\) to find the value of \(\mathrm{k}\)
  2. Then use that \(\mathrm{k}\) value as input to function \(\mathrm{q}\)
• For piecewise functions, the key is determining which piece to use based on the input value

3. TRANSLATE which piece of p(t) to use

• We need \(\mathrm{p(-3)}\), so we check: Is \(\mathrm{-3 \lt 0}\) or \(\mathrm{-3 ≥ 0}\)?
• Since \(\mathrm{-3 \lt 0}\), we use the first piece: \(\mathrm{p(t) = \frac{t + 7}{2}}\)

4. SIMPLIFY to find k = p(-3)

• Substitute \(\mathrm{t = -3}\) into \(\mathrm{\frac{t + 7}{2}}\):

\(\mathrm{p(-3) = \frac{-3 + 7}{2}}\)
\(\mathrm{p(-3) = \frac{4}{2}}\)
\(\mathrm{p(-3) = 2}\)

• Therefore: \(\mathrm{k = 2}\)

5. SIMPLIFY to find q(k) = q(2)

• Substitute \(\mathrm{s = 2}\) into \(\mathrm{q(s) = |5 - 2s| + s^2}\):

\(\mathrm{q(2) = |5 - 2(2)| + (2)^2}\)
\(\mathrm{q(2) = |5 - 4| + 4}\)
\(\mathrm{q(2) = |1| + 4}\)
\(\mathrm{q(2) = 1 + 4}\)
\(\mathrm{q(2) = 5}\)

Answer: B) 5

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misread the piecewise function conditions and use the wrong piece for \(\mathrm{p(-3)}\).

Many students see that \(\mathrm{-3}\) is negative but mistakenly think they should use the piece where \(\mathrm{t ≥ 0}\), or they get confused about which condition applies. They might calculate \(\mathrm{p(-3) = 3(-3) - 1 = -10}\), giving \(\mathrm{k = -10}\). Then \(\mathrm{q(-10) = |5 - 2(-10)| + (-10)^2 = |25| + 100 = 125}\).

This may lead them to select Choice D (125).

Second Most Common Error:

Incomplete INFER reasoning: Students correctly find \(\mathrm{k = 2}\) but then stop, thinking that's the final answer.

They successfully evaluate \(\mathrm{p(-3) = 2}\) but forget that the question asks for \(\mathrm{q(k)}\), not just \(\mathrm{k}\). They see \(\mathrm{k = 2}\) and look for 2 among the answer choices, but since 2 isn't listed, this leads to confusion and guessing.

The Bottom Line:

This problem tests whether students can handle the sequential nature of function composition while correctly interpreting piecewise function notation. The key insight is recognizing that finding \(\mathrm{k}\) is only the first step—the real answer comes from the second function evaluation.