Question:\(\mathrm{h(x) = 6(3)^x}\)The function h is defined by the given equation. If \(\mathrm{j(x) = h(x - 1)}\), which of the...

1. TRANSLATE the function relationship

• Given information:
  • \(\mathrm{h(x) = 6(3)^x}\)
  • \(\mathrm{j(x) = h(x - 1)}\)
• What this means: Wherever we see x in h(x), we replace it with (x - 1)

2. TRANSLATE into mathematical form

• Substitute (x - 1) into h(x):
  \(\mathrm{j(x) = h(x - 1) = 6(3)^{(x-1)}}\)
• This gives us \(\mathrm{j(x) = 6(3)^{(x-1)}}\)

3. SIMPLIFY using exponent rules

• Apply the rule \(\mathrm{a^{(m-n)} = a^m \times a^{(-n)}}\):
  \(\mathrm{6(3)^{(x-1)} = 6 \times 3^x \times 3^{(-1)}}\)
• Since \(\mathrm{3^{(-1)} = \frac{1}{3}}\):
  \(\mathrm{= 6 \times 3^x \times \left(\frac{1}{3}\right)}\)
• Multiply the constants:
  \(\mathrm{= \left(\frac{6}{3}\right) \times 3^x = 2 \times 3^x = 2(3)^x}\)

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students correctly get to \(\mathrm{j(x) = 6(3)^{(x-1)}}\) but then struggle with the exponent rules. They might try to "distribute" the exponent: \(\mathrm{6(3)^{(x-1)} = 6(3^x - 3^1) = 6(3^x - 3) = 6(3^x) - 18}\), leading to confusion since this doesn't match any answer choice format. This leads to guessing among the choices.

Second Most Common Error:

Incomplete TRANSLATE reasoning: Some students see \(\mathrm{j(x) = h(x-1)}\) and think it means "shift the coefficient" rather than "substitute (x-1) for x." They might incorrectly reason that since \(\mathrm{h(x) = 6(3)^x}\), then j(x) should be \(\mathrm{6(3)^x}\) multiplied by 3 (thinking about horizontal shifts incorrectly), leading them toward Choice B (\(\mathrm{18(3)^x}\)).

The Bottom Line:

This problem tests whether students can properly execute function substitution followed by careful algebraic simplification. The key insight is that \(\mathrm{j(x) = h(x-1)}\) means literal substitution, not a transformation of the function's graph properties.