Function h is defined by \(\mathrm{h(x) = (x - 6)(x - 2)(x + 1)}\). Function k is defined by \(\mathrm{k(x)...

1. TRANSLATE the function composition

• Given information:
  • \(\mathrm{h(x) = (x - 6)(x - 2)(x + 1)}\)
  • \(\mathrm{k(x) = h(-x)}\)
  • Need x-intercepts of \(\mathrm{k(x)}\)

• To find \(\mathrm{k(x)}\), substitute \(\mathrm{-x}\) for every \(\mathrm{x}\) in \(\mathrm{h(x)}\):
  \(\mathrm{k(x) = h(-x) = ((-x) - 6)((-x) - 2)((-x) + 1)}\)
  \(\mathrm{k(x) = (-x - 6)(-x - 2)(-x + 1)}\)

2. INFER the strategy for finding x-intercepts

• X-intercepts occur where the function equals zero
• Set \(\mathrm{k(x) = 0}\) and use zero product property

3. SIMPLIFY by applying zero product property

• Set \(\mathrm{k(x) = 0}\):
  \(\mathrm{(-x - 6)(-x - 2)(-x + 1) = 0}\)

• Each factor can equal zero:
  \(\mathrm{-x - 6 = 0 \rightarrow x = -6}\)
  \(\mathrm{-x - 2 = 0 \rightarrow x = -2}\)
  \(\mathrm{-x + 1 = 0 \rightarrow x = 1}\)

4. SIMPLIFY to find the final answer

• Sum the x-intercepts:
  \(\mathrm{p + q + r = (-6) + (-2) + 1 = -7}\)

Answer: (A) -7

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students incorrectly substitute \(\mathrm{-x}\) into the original function, often just putting a negative sign in front: \(\mathrm{k(x) = -(x - 6)(x - 2)(x + 1)}\) instead of properly substituting \(\mathrm{-x}\) for each \(\mathrm{x}\).

This leads to finding x-intercepts of the wrong function, typically getting \(\mathrm{x = 6, 2, -1}\), which sums to 7, leading them to select Choice (D) (7).

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{k(x) = (-x - 6)(-x - 2)(-x + 1) = 0}\) but make sign errors when solving the linear equations, such as solving \(\mathrm{-x - 6 = 0}\) as \(\mathrm{x = 6}\) instead of \(\mathrm{x = -6}\).

This results in incorrect x-intercepts and an incorrect sum, causing confusion and potentially leading to guessing among the remaining choices.

The Bottom Line:

This problem tests whether students can carefully execute function composition with negative inputs and maintain algebraic precision throughout. The key insight is that \(\mathrm{k(x) = h(-x)}\) requires complete substitution of \(\mathrm{-x}\) for \(\mathrm{x}\), not just adding a negative sign to the function.