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

1. TRANSLATE the problem requirements

• Given: Two exponential functions with domain \(\mathrm{x ≥ 0}\)
• Find: Which equation shows its minimum value as a visible constant or coefficient
• Key insight: The minimum value must appear explicitly in the equation, not require calculation

2. INFER the approach for exponential functions

• For \(\mathrm{f(x) = ab^x}\) where \(\mathrm{b \gt 1}\), minimum occurs at \(\mathrm{x = 0}\) with value a
• Since our functions have shifted exponents \(\mathrm{(x+2)}\) and \(\mathrm{(x+4)}\), we need to rewrite them
• The question asks if the minimum appears as a constant or coefficient in the given form

3. SIMPLIFY Function I: \(\mathrm{g(x) = 18(1.16)(1.4)^{(x+2)}}\)

• At minimum \(\mathrm{(x = 0)}\): \(\mathrm{g(0) = 18(1.16)(1.4)^2}\)
• Rewrite: \(\mathrm{g(x) = 18(1.16)(1.4)^2 × (1.4)^x}\)
• The minimum value \(\mathrm{18(1.16)(1.4)^2}\) is NOT visible in the original equation

4. SIMPLIFY Function II: \(\mathrm{h(x) = 18(1.4)^{(x+4)}}\)

• At minimum \(\mathrm{(x = 0)}\): \(\mathrm{h(0) = 18(1.4)^4}\)
• Rewrite: \(\mathrm{h(x) = 18(1.4)^4 × (1.4)^x}\)
• The minimum value \(\mathrm{18(1.4)^4}\) is NOT visible in the original equation

5. INFER the final conclusion

• Neither equation displays its minimum value as an explicit constant or coefficient
• Both require calculation to determine the minimum value

Answer: D. Neither I nor II

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER reasoning: Students calculate the minimum values correctly but misunderstand what "displays as a constant or coefficient" means. They think that because they can calculate \(\mathrm{g(0) = 18(1.16)(1.4)^2}\) and \(\mathrm{h(0) = 18(1.4)^4}\), both functions display their minimums. They miss that the question asks whether these minimum values are visible in the given equations without computation.

This may lead them to select Choice C (I and II).

Second Most Common Error:

Inadequate SIMPLIFY execution: Students attempt to rewrite the functions but make algebraic errors or don't complete the transformation to standard form. They might incorrectly identify 18 or 18(1.16) as minimum values without accounting for the exponential shifts.

This may lead them to select Choice A (I only) or Choice B (II only).

The Bottom Line:

The key challenge is distinguishing between values that can be calculated from an equation versus values that are explicitly displayed in the equation's given form. This requires careful interpretation of mathematical language.