Question:\(\mathrm{h(t) = 8(9)^t}\)The function h is defined by the given equation. If \(\mathrm{k(t) = h(t + 2)}\), which of the...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{h(t) = 8(9)^t}\)
  • \(\mathrm{k(t) = h(t + 2)}\)
• What this tells us: We need to find an explicit equation for k by substituting into h

2. TRANSLATE the composition notation

• The notation \(\mathrm{k(t) = h(t + 2)}\) means we replace every t in h(t) with (t + 2)
• This gives us: \(\mathrm{k(t) = 8(9)^{(t + 2)}}\)

3. SIMPLIFY using exponent rules

• Apply the rule \(\mathrm{a^{(m + n)} = a^m \cdot a^n}\) to break apart the exponent:
  \(\mathrm{k(t) = 8(9)^t \cdot (9)^2}\)
• Calculate \(\mathrm{(9)^2 = 81}\):
  \(\mathrm{k(t) = 8(9)^t \cdot 81}\)

4. SIMPLIFY by multiplying coefficients

• Multiply \(\mathrm{8 \times 81 = 648}\) (use calculator if needed):
  \(\mathrm{k(t) = 648(9)^t}\)

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret what \(\mathrm{k(t) = h(t + 2)}\) means and try to add 2 to the entire function instead of substituting (t + 2) for t.

They might write \(\mathrm{k(t) = 8(9)^t + 2}\) or \(\mathrm{k(t) = 8(9)^t + 2}\), completely missing the function composition concept. This leads to confusion and guessing since none of the answer choices match this approach.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{k(t) = 8(9)^{(t+2)}}\) but make arithmetic errors when calculating \(\mathrm{9^2}\) or \(\mathrm{8 \times 81}\).

If they calculate \(\mathrm{9^2 = 18}\) (confusing with 9 × 2), they get \(\mathrm{k(t) = 8(9)^t \cdot 18 = 144(9)^t}\), which doesn't match any answer choice. If they miscalculate \(\mathrm{8 \times 81}\) as something other than 648, they might select a wrong answer or get stuck.

The Bottom Line:

This problem requires recognizing that function composition means input substitution, not addition to the output, combined with careful application of exponent rules and arithmetic.