For x gt 0, the function g is defined as follows: \(\mathrm{g(x)}\) equals the total cost when a 15% tax...

1. TRANSLATE the problem information

• Given information:
  • g(x) = total cost when 15% tax is added to base price x
  • We need to determine if this function is increasing/decreasing and linear/exponential

• What this tells us: We need to write g(x) as a mathematical expression first

2. TRANSLATE the tax calculation into mathematics

• Base price: x
• Tax amount: \(15\% \text{ of } \mathrm{x} = 0.15\mathrm{x}\)
• Total cost: base price + tax = \(\mathrm{x} + 0.15\mathrm{x} = 1.15\mathrm{x}\)

Therefore: \(\mathrm{g(x)} = 1.15\mathrm{x}\)

3. INFER the function type

• The function \(\mathrm{g(x)} = 1.15\mathrm{x}\) is in the form \(\mathrm{mx} + \mathrm{b}\) where:
  • \(\mathrm{m} = 1.15\) (coefficient)
  • \(\mathrm{b} = 0\) (y-intercept)
• This makes it a linear function
• Exponential functions have x in the exponent (like \(2^\mathrm{x}\)), not as the base

4. INFER whether increasing or decreasing

• Since the coefficient \(1.15 \gt 0\), when x increases, g(x) also increases
• This makes the function increasing

Answer: D (Increasing linear)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER reasoning about "increasing" vs "decreasing": Students think about tax from a consumer perspective - "tax is bad, so the function should be decreasing" or "tax reduces my purchasing power." However, mathematically, as the base price x gets larger, the total cost g(x) also gets larger, making it an increasing function.

This may lead them to select Choice B (Decreasing linear)

Second Most Common Error:

Poor TRANSLATE execution: Students might incorrectly set up the tax calculation, perhaps writing \(\mathrm{g(x)} = \mathrm{x} - 0.15\mathrm{x}\) (subtracting tax instead of adding) or \(\mathrm{g(x)} = 0.15\mathrm{x}\) (forgetting to include the base price).

This leads to confusion about whether the function is increasing or decreasing and causes guessing.

The Bottom Line:

This problem tests whether students can separate mathematical behavior from real-world intuition. While tax feels "negative" from a spending perspective, the mathematical function describing total cost still increases as base price increases.