Function f is defined by \(\mathrm{f(x) = (x + 6)(x + 5)(x + 1)}\). Function g is defined by \(\mathrm{g(x)...

1. TRANSLATE the function composition

• Given information:
  • \(\mathrm{f(x) = (x + 6)(x + 5)(x + 1)}\)
  • \(\mathrm{g(x) = f(x - 1)}\)
• What this means: Replace every x in f(x) with (x - 1)

2. SIMPLIFY the substitution

• \(\mathrm{g(x) = f(x - 1) = ((x - 1) + 6)((x - 1) + 5)((x - 1) + 1)}\)
• Simplify each factor carefully:
  • \(\mathrm{(x - 1) + 6 = x + 5}\)
  • \(\mathrm{(x - 1) + 5 = x + 4}\)
  • \(\mathrm{(x - 1) + 1 = x + 0 = x}\)
• Therefore: \(\mathrm{g(x) = x(x + 4)(x + 5)}\)

3. INFER what x-intercepts represent

• X-intercepts occur where the graph crosses the x-axis
• This means we need to solve \(\mathrm{g(x) = 0}\)

4. APPLY the zero product property

• Set up: \(\mathrm{x(x + 4)(x + 5) = 0}\)
• Since the product equals zero, at least one factor must equal zero:
  • \(\mathrm{x = 0}\) OR \(\mathrm{x + 4 = 0}\) OR \(\mathrm{x + 5 = 0}\)
• Solve each equation:
  • \(\mathrm{x = 0, x = -4, x = -5}\)

5. SIMPLIFY the final calculation

• The x-intercepts are \(\mathrm{(0, 0), (-4, 0),}\) and \(\mathrm{(-5, 0)}\)
• Therefore: \(\mathrm{a + b + c = 0 + (-4) + (-5) = -9}\)

Answer: B. -9

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse \(\mathrm{g(x) = f(x - 1)}\) with \(\mathrm{g(x) = f(x + 1)}\), substituting \(\mathrm{(x + 1)}\) instead of \(\mathrm{(x - 1)}\).

This leads to \(\mathrm{f(x + 1) = ((x + 1) + 6)((x + 1) + 5)((x + 1) + 1) = (x + 7)(x + 6)(x + 2)}\), giving x-intercepts of 0, -6, and -7. The sum would be -13, which doesn't match any answer choice, causing confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students make sign errors when simplifying \(\mathrm{(x - 1) + 6}\), getting results like \(\mathrm{(x - 7)}\) instead of \(\mathrm{(x + 5)}\).

For example, if they incorrectly get \(\mathrm{g(x) = (x - 7)(x - 6)(x - 2)}\), they'd find x-intercepts at 7, 6, and 2, leading to a sum of 15. This may lead them to select Choice D (15).

The Bottom Line:

Function composition with translations requires careful attention to signs and systematic algebraic substitution. The key insight is that \(\mathrm{g(x) = f(x - 1)}\) shifts every input by 1 unit, which affects where the roots occur.