If \(\mathrm{g(x) = x^2 - 6x}\) and \(\mathrm{h(x) = x + 2}\), what is \(\mathrm{g(h(-1))}\)?

1. TRANSLATE the problem notation

• Given information:
  • \(\mathrm{g(x) = x^2 - 6x}\)
  • \(\mathrm{h(x) = x + 2}\)
  • Need to find \(\mathrm{g(h(-1))}\)

• What this notation means: \(\mathrm{g(h(-1))}\) is a composition where we evaluate \(\mathrm{h(-1)}\) first, then use that result as the input for function g

2. INFER the solution strategy

• Function compositions work from the inside out
• We must evaluate the inner function \(\mathrm{h(-1)}\) before we can evaluate the outer function g
• Think of it as: \(\mathrm{g(h(-1)) = g(?)}\) where \(\mathrm{? = h(-1)}\)

3. SIMPLIFY by evaluating the inner function first

• Find \(\mathrm{h(-1)}\):
  \(\mathrm{h(x) = x + 2}\)
  \(\mathrm{h(-1) = (-1) + 2 = 1}\)

4. SIMPLIFY by evaluating the outer function with our result

• Now find \(\mathrm{g(1)}\):
  \(\mathrm{g(x) = x^2 - 6x}\)
  \(\mathrm{g(1) = (1)^2 - 6(1) = 1 - 6 = -5}\)

Answer: A. -5

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the inside-out evaluation order and try to substitute -1 directly into \(\mathrm{g(x)}\) instead of \(\mathrm{h(x)}\) first.

They might calculate \(\mathrm{g(-1) = (-1)^2 - 6(-1) = 1 + 6 = 7}\), then realize this isn't among the answer choices and get confused. This leads to guessing or selecting an answer that seems close.

Second Most Common Error:

Poor SIMPLIFY execution: Students understand the correct order but make arithmetic errors, particularly with negative numbers or when calculating \(\mathrm{g(1)}\).

For example, they might correctly find \(\mathrm{h(-1) = 1}\), but then calculate \(\mathrm{g(1) = 1^2 - 6(1) = 1 - 6 = 5}\) (forgetting the negative), leading them to select Choice D (5).

The Bottom Line:

Function composition problems require careful attention to order - always work from the innermost function outward, and double-check your arithmetic at each step.