The functions h and k are defined by the given equations.\(\mathrm{h(x) = 1 + |x^2 - 5x|}\)\(\mathrm{k(t) = |\frac{6}{t-1}| -...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{h(x) = 1 + |x² - 5x|}\)
  • \(\mathrm{k(t) = |6/(t-1)| - t + 8}\), where \(\mathrm{t ≠ 1}\)
  • \(\mathrm{h(6) = d}\) (we need to find d first)
  • Find \(\mathrm{k(d)}\)

• What this tells us: We have a two-step process - first evaluate h at x = 6, then use that result as input for function k.

2. SIMPLIFY to find h(6)

• Substitute x = 6 into h(x):

\(\mathrm{h(6) = 1 + |6² - 5(6)|}\)

• Calculate inside the absolute value first:

\(\mathrm{6² - 5(6) = 36 - 30 = 6}\)

• Apply absolute value: \(\mathrm{|6| = 6}\)

• Complete the calculation: \(\mathrm{h(6) = 1 + 6 = 7}\)

• Therefore: \(\mathrm{d = 7}\)

3. SIMPLIFY to find k(7)

• Substitute t = 7 into k(t):

\(\mathrm{k(7) = |6/(7-1)| - 7 + 8}\)

• Calculate the fraction: \(\mathrm{6/(7-1) = 6/6 = 1}\)

• Apply absolute value: \(\mathrm{|1| = 1}\)

• Complete the calculation: \(\mathrm{k(7) = 1 - 7 + 8 = 2}\)

Answer: 2

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors when evaluating \(\mathrm{6² - 5(6)}\), calculating it as \(\mathrm{6² - 5 × 6 = 36 - 5 × 6 = 36 - 11 = 25}\) instead of \(\mathrm{36 - 30 = 6}\).

This leads to \(\mathrm{h(6) = 1 + |25| = 26}\), so \(\mathrm{d = 26}\), then \(\mathrm{k(26)}\) requires evaluating k at an unintended value, causing confusion and likely guessing.

Second Most Common Error:

Poor TRANSLATE reasoning: Students misunderstand the two-step nature of the problem, trying to find \(\mathrm{k(6)}\) directly instead of first finding \(\mathrm{d = h(6)}\), then evaluating \(\mathrm{k(d)}\).

This causes them to substitute 6 into the k function immediately, getting \(\mathrm{k(6) = |6/(6-1)| - 6 + 8 = |6/5| - 6 + 8 = 1.2 + 2 = 3.2}\), leading to confusion since this doesn't match typical answer formats.

The Bottom Line:

This problem tests your ability to follow a multi-step process systematically while handling absolute values correctly. The key is recognizing that you must complete the first function evaluation entirely before moving to the second function.