The function f is defined by \(\mathrm{f(x) = x^2 - 3x + k}\), where k is a constant. If \(\mathrm{f(2)...

1. TRANSLATE the problem information

• Given information:
  • Function: \(\mathrm{f(x) = x^2 - 3x + k}\) (where k is unknown)
  • \(\mathrm{f(2) = 8}\)
  • \(\mathrm{f(-1) = 14}\)
• Need to find: \(\mathrm{f(1) × f(5)}\)

2. INFER the solution strategy

• Key insight: We can't evaluate f(1) and f(5) until we know the value of k
• Strategy: Use the given function values to find k first

3. SIMPLIFY to find the constant k

• Using \(\mathrm{f(2) = 8}\):

\(\mathrm{f(2) = (2)^2 - 3(2) + k}\)
\(\mathrm{= 4 - 6 + k}\)
\(\mathrm{= k - 2}\)

• Since \(\mathrm{f(2) = 8}\): \(\mathrm{k - 2 = 8}\)
• Therefore: \(\mathrm{k = 10}\)

4. Verify our value of k

• Check with \(\mathrm{f(-1) = 14}\):

\(\mathrm{f(-1) = (-1)^2 - 3(-1) + 10}\)
\(\mathrm{= 1 + 3 + 10}\)
\(\mathrm{= 14}\) ✓

5. SIMPLIFY to find the required function values

• Now that \(\mathrm{f(x) = x^2 - 3x + 10}\):

\(\mathrm{f(1) = (1)^2 - 3(1) + 10}\)
\(\mathrm{= 1 - 3 + 10}\)
\(\mathrm{= 8}\)

\(\mathrm{f(5) = (5)^2 - 3(5) + 10}\)
\(\mathrm{= 25 - 15 + 10}\)
\(\mathrm{= 20}\)

6. Calculate the final product

\(\mathrm{f(1) × f(5) = 8 × 20 = 160}\)

Answer: C) 160

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students try to evaluate f(1) and f(5) without first determining k, realizing they can't proceed without this unknown constant.

This leads to confusion and guessing, or students may incorrectly assume \(\mathrm{k = 0}\) and get \(\mathrm{f(1) × f(5) = (-2) × 10 = -20}\), which doesn't match any answer choice.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify they need to find k but make calculation errors. For example, when solving \(\mathrm{k - 2 = 8}\), they might get \(\mathrm{k = 6}\) instead of \(\mathrm{k = 10}\).

With \(\mathrm{k = 6}\), they would get \(\mathrm{f(1) = 4}\) and \(\mathrm{f(5) = 16}\), leading to \(\mathrm{f(1) × f(5) = 64}\), which corresponds to Choice A (64).

The Bottom Line:

This problem tests whether students can work systematically with unknown parameters in functions, requiring them to use given information strategically before proceeding to the main calculation.