For p gt 0, the value q is determined such that p is 125% of q. The function g is...

1. TRANSLATE the relationship into mathematics

• Given information:
  • \(\mathrm{p \gt 0}\) (p is positive)
  • "p is 125% of q"
  • Function g is defined by \(\mathrm{g(p) = q}\)

• TRANSLATE the key phrase: "p is 125% of q" means \(\mathrm{p = 1.25q}\)

2. INFER what the function definition requires

• The function \(\mathrm{g(p) = q}\) means:
  • Input: p
  • Output: q
  • We need q expressed in terms of p, not the other way around

• Strategy: Solve the equation \(\mathrm{p = 1.25q}\) for q

3. SIMPLIFY to find the function

• From \(\mathrm{p = 1.25q}\), solve for q:
  • \(\mathrm{q = p/1.25}\)
  • Convert to fraction: \(\mathrm{1.25 = 5/4}\)
  • \(\mathrm{q = p ÷ (5/4)}\)
    \(\mathrm{= p × (4/5)}\)
    \(\mathrm{= 0.8p}\)

• Therefore: \(\mathrm{g(p) = 0.8p}\)

4. INFER the function type and behavior

• The function \(\mathrm{g(p) = 0.8p}\) is in the form \(\mathrm{y = mx}\) where \(\mathrm{m = 0.8}\)
  • This is the definition of a linear function
  • Since \(\mathrm{m = 0.8 \gt 0}\), the slope is positive
  • Positive slope means the function is increasing

Answer: D. Increasing linear

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may incorrectly interpret "p is 125% of q" as "q is 125% of p," leading to \(\mathrm{q = 1.25p}\) instead of \(\mathrm{p = 1.25q}\).

If they use \(\mathrm{q = 1.25p}\), they get \(\mathrm{g(p) = 1.25p}\), which is still increasing linear, so they might still select the correct answer by accident. However, the more common error is setting up \(\mathrm{q = p/1.25}\) incorrectly as \(\mathrm{q = 1.25/p}\), leading to an inverse relationship.

This may lead them to select Choice A (Decreasing exponential) or Choice B (Decreasing linear).

Second Most Common Error:

Poor INFER reasoning: Students may not recognize that they need to express q in terms of p. They might try to work with \(\mathrm{p = 1.25q}\) directly and get confused about which variable should be the input and which should be the output.

This leads to confusion and guessing among the answer choices.

The Bottom Line:

This problem tests whether students can accurately translate percentage relationships into equations and then manipulate those equations to express functions correctly. The key insight is recognizing that "A is 125% of B" means \(\mathrm{A = 1.25B}\), not \(\mathrm{B = 1.25A}\).