The function p is defined by the equation \(\mathrm{p(x) = 81(3)^x}\). The function q is related to p such that...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{p(x) = 81(3)^x}\)
  • \(\mathrm{p(x) = q(x + 2)}\)
• We need to find the equation for \(\mathrm{q(x)}\)

2. INFER the relationship between functions

• The key insight: if \(\mathrm{p(x) = q(x + 2)}\), then we need to "work backwards"
• To find \(\mathrm{q(x)}\), we need to determine what input to p gives us \(\mathrm{q(x)}\)
• Since \(\mathrm{p(x) = q(x + 2)}\), then \(\mathrm{p(x - 2) = q(x)}\)
• This means: \(\mathrm{q(x) = p(x - 2)}\)

3. TRANSLATE this relationship into the equation

• Substitute \(\mathrm{(x - 2)}\) into the definition of \(\mathrm{p(x)}\):
• \(\mathrm{q(x) = p(x - 2) = 81(3)^{(x - 2)}}\)

4. SIMPLIFY the exponential expression

• Use the exponent property: \(\mathrm{a^{(b - c)} = a^b / a^c}\)
• So \(\mathrm{3^{(x - 2)} = 3^x / 3^2 = 3^x / 9}\)
• Therefore: \(\mathrm{q(x) = 81 \times (3^x / 9)}\)

5. SIMPLIFY the coefficient

• \(\mathrm{q(x) = (81/9) \times 3^x = 9 \times 3^x = 9(3)^x}\)

Answer: A. \(\mathrm{q(x) = 9(3)^x}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students struggle to determine the correct relationship between p and q from the condition \(\mathrm{p(x) = q(x + 2)}\). They might incorrectly think that \(\mathrm{q(x) = p(x + 2)}\) instead of \(\mathrm{q(x) = p(x - 2)}\), or they might try to directly manipulate the equation without establishing this fundamental relationship first.

This leads to confusion and guessing, or potentially selecting an incorrect answer choice by working with the wrong relationship.

Second Most Common Error:

Poor SIMPLIFY execution: Even if students correctly identify that \(\mathrm{q(x) = p(x - 2)}\), they make algebraic errors when simplifying \(\mathrm{81(3)^{(x-2)}}\). Common mistakes include incorrectly applying exponent rules or making arithmetic errors when computing \(\mathrm{81/9}\).

This may lead them to select Choice B (\(\mathrm{q(x) = 27(3)^x}\)) or Choice C (\(\mathrm{q(x) = 81(9)^x}\)) depending on the specific computational error.

The Bottom Line:

This problem requires students to think "backwards" from a function composition relationship, which is conceptually challenging. The algebraic manipulation is straightforward once the relationship is established, but identifying that relationship requires solid reasoning about function transformations.