The functions f and g are defined by the given equations, where x geq 0. Which of the following equations...

1. TRANSLATE the problem requirements

• Given information:
  • \(\mathrm{f(x) = 33(0.4)^{(x+3)}}\) where \(\mathrm{x \geq 0}\)
  • \(\mathrm{g(x) = 33(0.16)(0.4)^{(x-2)}}\) where \(\mathrm{x \geq 0}\)
  • Need to find which equation shows its maximum value as a constant or coefficient

2. INFER the behavior of these functions

• Since both functions have base 0.4, and \(\mathrm{0.4 \lt 1}\), these are decreasing exponential functions
• Decreasing functions achieve their maximum value at the leftmost point of their domain
• Since \(\mathrm{x \geq 0}\), the maximum occurs at \(\mathrm{x = 0}\)

3. SIMPLIFY to find the maximum value of f(x)

• Calculate \(\mathrm{f(0)}\): \(\mathrm{f(0) = 33(0.4)^{(0+3)}}\)
• \(\mathrm{f(0) = 33(0.4)^3}\)
• Compute: \(\mathrm{(0.4)^3 = 0.064}\) (use calculator)
• So \(\mathrm{f(0) = 33(0.064) = 2.112}\) (use calculator)
• The maximum value of f is 2.112

4. INFER whether f(x) displays this maximum

• Look at the equation: \(\mathrm{f(x) = 33(0.4)^{(x+3)}}\)
• The visible constants/coefficients are 33 and 0.4
• Neither equals 2.112, so f(x) does NOT display its maximum value

5. SIMPLIFY to find the maximum value of g(x)

• Calculate \(\mathrm{g(0)}\): \(\mathrm{g(0) = 33(0.16)(0.4)^{(0-2)}}\)
• \(\mathrm{g(0) = 33(0.16)(0.4)^{(-2)}}\)
• Use exponential rule: \(\mathrm{(0.4)^{(-2)} = \frac{1}{(0.4)^2}}\)
• \(\mathrm{\frac{1}{(0.4)^2} = \frac{1}{0.16} = 6.25}\) (use calculator)
• So \(\mathrm{g(0) = 33(0.16)(6.25)}\)
• \(\mathrm{g(0) = 33(0.16 \times 6.25)}\)
• \(\mathrm{g(0) = 33(1) = 33}\)
• The maximum value of g is 33

6. INFER whether g(x) displays this maximum

• Look at the equation: \(\mathrm{g(x) = 33(0.16)(0.4)^{(x-2)}}\)
• The visible constants/coefficients are 33, 0.16, and 0.4
• The coefficient 33 equals the maximum value 33, so g(x) DOES display its maximum value

Answer: B. II only

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students correctly calculate both maximum values but misunderstand what "displays as a constant or coefficient" means. They think the question asks which function HAS a maximum value that can be calculated, rather than which equation literally SHOWS that maximum as a visible term.

This leads them to think both equations "work" since both have calculable maximums, causing them to select Choice C (I and II).

Second Most Common Error:

Poor SIMPLIFY execution: Students make computational errors when handling the negative exponent in \(\mathrm{g(0)}\), either forgetting the rule \(\mathrm{a^{(-n)} = \frac{1}{a^n}}\) or making arithmetic mistakes with \(\mathrm{\frac{1}{(0.4)^2}}\).

This prevents them from finding \(\mathrm{g(0) = 33}\), leading to confusion about whether g(x) displays its maximum. They may abandon systematic solution and guess among the choices.

The Bottom Line:

This problem tests both computational skills with exponential expressions AND reading comprehension about what mathematical "display" means - a combination that catches many students off guard.