The function f is defined by \(\mathrm{f(x) = (x - 3)(x + 2)}\). Which of the following is a positive...

1. TRANSLATE the problem requirements

• Given: \(\mathrm{f(x) = (x - 3)(x + 2)}\)
• Find: The positive x-intercept of \(\mathrm{y = f(x)}\)
• Key insight: X-intercepts occur where the graph crosses the x-axis, meaning \(\mathrm{y = 0}\), so \(\mathrm{f(x) = 0}\)

2. INFER the solution approach

• Since \(\mathrm{f(x)}\) is already factored, we can use the zero product property
• Set the function equal to zero: \(\mathrm{(x - 3)(x + 2) = 0}\)

3. INFER and apply the zero product property

• If \(\mathrm{(x - 3)(x + 2) = 0}\), then either:
  • \(\mathrm{x - 3 = 0}\), which gives \(\mathrm{x = 3}\)
  • \(\mathrm{x + 2 = 0}\), which gives \(\mathrm{x = -2}\)
• Both solutions are x-intercepts: \(\mathrm{x = 3}\) and \(\mathrm{x = -2}\)

4. APPLY CONSTRAINTS to select the correct answer

• The question asks specifically for the positive x-intercept
• Between \(\mathrm{x = 3}\) and \(\mathrm{x = -2}\), only \(\mathrm{x = 3}\) is positive
• Looking at the answer choices, (D) 3 matches our positive solution

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may not understand that "x-intercept" means setting \(\mathrm{f(x) = 0}\).

Some students think x-intercept means "substitute x-values from the choices" or get confused about what the intercept represents. They might try plugging in answer choices randomly or attempt to find where the function equals some other value. This leads to confusion and guessing.

Second Most Common Error:

Poor APPLY CONSTRAINTS reasoning: Students solve correctly to find both \(\mathrm{x = 3}\) and \(\mathrm{x = -2}\), but then select the negative value.

They correctly find both solutions but fail to focus on the word "positive" in the question. Seeing -2 as an answer choice (A), they might select it without checking the constraint. This leads them to select Choice A (-2).

The Bottom Line:

This problem tests whether students understand the fundamental connection between x-intercepts and zeros of a function, plus their ability to apply given constraints to select from multiple valid solutions.