Function f is defined by \(\mathrm{f(x) = (x - 4)(x - 1)(x + 3)}\). Function g is defined by \(\mathrm{g(x)...

1. TRANSLATE the function composition

• Given information:
  • \(\mathrm{f(x) = (x - 4)(x - 1)(x + 3)}\)
  • \(\mathrm{g(x) = f(2 - x)}\)
• This means: substitute \(\mathrm{(2 - x)}\) everywhere you see \(\mathrm{x}\) in \(\mathrm{f(x)}\)

2. SIMPLIFY the substitution

• Replace \(\mathrm{x}\) with \(\mathrm{(2 - x)}\) in \(\mathrm{f(x)}\):

\(\mathrm{g(x) = ((2 - x) - 4)((2 - x) - 1)((2 - x) + 3)}\)

• Simplify each factor:
  • \(\mathrm{(2 - x) - 4 = -2 - x}\)
  • \(\mathrm{(2 - x) - 1 = 1 - x}\)
  • \(\mathrm{(2 - x) + 3 = 5 - x}\)
• Therefore: \(\mathrm{g(x) = (-2 - x)(1 - x)(5 - x)}\)

3. INFER how to find x-intercepts

• X-intercepts occur where the graph crosses the x-axis
• This happens when \(\mathrm{y = 0}\), so we need \(\mathrm{g(x) = 0}\)
• Set the factored form equal to zero: \(\mathrm{(-2 - x)(1 - x)(5 - x) = 0}\)

4. APPLY CONSTRAINTS using zero product property

• If a product equals zero, at least one factor must be zero:
  • \(\mathrm{-2 - x = 0}\) → \(\mathrm{x = -2}\)
  • \(\mathrm{1 - x = 0}\) → \(\mathrm{x = 1}\)
  • \(\mathrm{5 - x = 0}\) → \(\mathrm{x = 5}\)

5. SIMPLIFY the final calculation

• Sum the x-intercepts: \(\mathrm{a + b + c = -2 + 1 + 5 = 4}\)

Answer: C (4)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may incorrectly substitute \(\mathrm{(2 - x)}\) into \(\mathrm{f(x)}\), creating sign errors or forgetting to distribute the substitution to all parts of the function. For example, they might write \(\mathrm{g(x) = (2 - x - 4)(2 - x - 1)(2 - x + 3) = (-2 + x)(1 + x)(5 + x)}\) by making sign errors during simplification. This leads to finding x-intercepts at \(\mathrm{x = 2}\), \(\mathrm{x = -1}\), and \(\mathrm{x = -5}\), giving a sum of -4, which may lead them to select Choice A (-4).

Second Most Common Error:

Conceptual confusion about x-intercepts: Some students may find the zeros of the original function \(\mathrm{f(x)}\) instead of \(\mathrm{g(x)}\), thinking that function composition doesn't change the x-intercepts. The original \(\mathrm{f(x) = (x - 4)(x - 1)(x + 3)}\) has zeros at \(\mathrm{x = 4}\), \(1\), \(\mathrm{-3}\), giving a sum of 2. This may lead them to select Choice B (2).

The Bottom Line:

This problem tests both algebraic manipulation skills and conceptual understanding of function composition. Students must carefully track substitutions and remember that transforming a function changes its intercepts.