If \(\mathrm{f(x) = \frac{x^2 - 6x + 3}{x - 1}}\), what is \(\mathrm{f(-1)}\)?

1. TRANSLATE the problem information

• Given: \(\mathrm{f(x) = \frac{x^2 - 6x + 3}{x - 1}}\)
• Need to find: \(\mathrm{f(-1)}\)
• This means substitute \(\mathrm{x = -1}\) into the function definition

2. TRANSLATE the substitution

• Replace every x with (-1):

\(\mathrm{f(-1) = \frac{(-1)^2 - 6(-1) + 3}{(-1) - 1}}\)

3. SIMPLIFY the numerator step by step

• Start with: \(\mathrm{(-1)^2 - 6(-1) + 3}\)
• Evaluate the exponent first: \(\mathrm{(-1)^2 = 1}\)
• Now we have: \(\mathrm{1 - 6(-1) + 3}\)
• Handle the multiplication: \(\mathrm{-6(-1) = +6}\)
• Now we have: \(\mathrm{1 + 6 + 3 = 10}\)

4. SIMPLIFY the denominator

• \(\mathrm{(-1) - 1 = -2}\)

5. SIMPLIFY the final fraction

• \(\mathrm{f(-1) = \frac{10}{-2} = -5}\)

Answer: A. -5

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill with negative numbers: Students often make sign errors when working with negative numbers, particularly confusing \(\mathrm{(-1)^2}\) with \(\mathrm{-1^2}\), or incorrectly computing \(\mathrm{-6(-1)}\) as \(\mathrm{-6}\) instead of \(\mathrm{+6}\).

For example, if they incorrectly think \(\mathrm{-6(-1) = -6}\), their numerator becomes \(\mathrm{1 - 6 + 3 = -2}\), giving them \(\mathrm{f(-1) = \frac{-2}{-2} = 1}\), which isn't even among the choices. This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution with fraction signs: Students correctly compute the numerator as 10 but make an error with the denominator's sign, treating \(\mathrm{(-1) - 1}\) as positive 2 instead of -2.

This gives them \(\mathrm{f(-1) = \frac{10}{2} = 5}\), leading them to select Choice D (5).

The Bottom Line:

This problem tests careful attention to signs and order of operations with negative numbers. The key challenge is maintaining accuracy through multiple steps involving negative number arithmetic, where small sign errors compound to produce completely wrong answers.