\(\mathrm{f(x) = 9(4)^x}\)The function f is defined by the given equation. If \(\mathrm{g(x) = f(x + 2)}\), which of the...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{f(x) = 9(4)^x}\)
  • \(\mathrm{g(x) = f(x + 2)}\)
• What this tells us: We need to substitute \(\mathrm{(x + 2)}\) wherever we see \(\mathrm{x}\) in the function \(\mathrm{f(x)}\)

2. TRANSLATE the function composition into substitution

• Since \(\mathrm{g(x) = f(x + 2)}\), we replace every \(\mathrm{x}\) in \(\mathrm{f(x)}\) with \(\mathrm{(x + 2)}\)
• \(\mathrm{f(x + 2) = 9(4)^{(x+2)}}\)

3. SIMPLIFY using exponent properties

• Apply the property: \(\mathrm{a^{(m+n)} = a^m \times a^n}\)
• \(\mathrm{(4)^{(x+2)} = (4)^x \times (4)^2}\)
• So \(\mathrm{f(x + 2) = 9 \times (4)^x \times (4)^2}\)

4. SIMPLIFY the arithmetic

• Calculate \(\mathrm{(4)^2 = 16}\)
• \(\mathrm{f(x + 2) = 9 \times (4)^x \times 16 = 144(4)^x}\)
• Therefore: \(\mathrm{g(x) = 144(4)^x}\)

Answer: B. \(\mathrm{g(x) = 144(4)^x}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may not understand that "\(\mathrm{g(x) = f(x + 2)}\)" means to substitute \(\mathrm{(x + 2)}\) for every \(\mathrm{x}\) in the original function. They might try to add 2 to the entire function or manipulate it in other incorrect ways.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{f(x + 2) = 9(4)^{(x+2)}}\) but then incorrectly apply exponent properties. They might think \(\mathrm{(4)^{(x+2)} = (4)^x + (4)^2}\) instead of \(\mathrm{(4)^x \times (4)^2}\), or make arithmetic errors when calculating \(\mathrm{4^2}\) or multiplying by 9.

This may lead them to select Choice A (\(\mathrm{18(4)^x}\)) if they calculated \(\mathrm{4^2 = 2}\), or other incorrect choices.

The Bottom Line:

This problem tests whether students understand function composition notation and can correctly apply exponent properties. The key insight is recognizing that function composition requires complete substitution, not just adding values to the function.