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

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{f(x) = 18(1.25)^x + 41}\) where \(\mathrm{x \geq 0}\)
  • \(\mathrm{g(x) = 9(0.73)^x}\) where \(\mathrm{x \geq 0}\)
• Need to find: Which equation shows its maximum value as a constant or coefficient

2. INFER the behavior of each exponential function

• For \(\mathrm{f(x) = 18(1.25)^x + 41}\):
  • Base \(\mathrm{1.25 \gt 1}\), so \(\mathrm{(1.25)^x}\) increases as x increases
  • This means \(\mathrm{f(x)}\) is increasing for \(\mathrm{x \geq 0}\)
  • Increasing functions have no maximum value on \(\mathrm{[0, \infty)}\)
• For \(\mathrm{g(x) = 9(0.73)^x}\):
  • Base \(\mathrm{0.73 \lt 1}\), so \(\mathrm{(0.73)^x}\) decreases as x increases
  • This means \(\mathrm{g(x)}\) is decreasing for \(\mathrm{x \geq 0}\)
  • Decreasing functions reach maximum at left endpoint

3. INFER where the maximum occurs and calculate it

• Since \(\mathrm{g(x)}\) is decreasing, maximum occurs at \(\mathrm{x = 0}\):
  \(\mathrm{g(0) = 9(0.73)^0}\)
  \(\mathrm{= 9(1)}\)
  \(\mathrm{= 9}\)

4. INFER which equation displays this maximum as a coefficient

• Function g has maximum value 9
• In the equation \(\mathrm{g(x) = 9(0.73)^x}\), the number 9 appears as the coefficient
• Function f has no maximum, so this condition isn't met

Answer: B. II only

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing that exponential function behavior depends on the base value

Students might think all exponential functions behave the same way, or they might not remember that when the base is between 0 and 1, the function decreases. They could incorrectly conclude that both functions have maximum values, or that neither has a maximum.

This may lead them to select Choice C (I and II) or Choice D (Neither I nor II)

Second Most Common Error:

Conceptual confusion about maximum identification: Students find the correct maximum value \(\mathrm{g(0) = 9}\) but don't recognize that this appears as the coefficient in the original equation

They might look for the number 9 to appear as a separate constant term (like the +41 in function I) rather than recognizing it as the coefficient of the exponential term.

This leads to confusion and guessing among the answer choices.

The Bottom Line:

This problem tests understanding of how exponential function behavior changes based on the base value, combined with the ability to connect maximum values back to the coefficients and constants in the original equation.