The function f is defined by \(\mathrm{f(x) = 4x + k(x - 1)}\), where k is a constant, and \(\mathrm{f(5)...

1. TRANSLATE the given condition into an equation

• Given information:
  • \(\mathrm{f(x) = 4x + k(x - 1)}\) where \(\mathrm{k}\) is unknown
  • \(\mathrm{f(5) = 32}\)
  • Need to find \(\mathrm{f(10)}\)

• TRANSLATE \(\mathrm{f(5) = 32}\) means: when \(\mathrm{x = 5}\), the function value is 32

2. INFER the solution strategy

• Key insight: We can't find \(\mathrm{f(10)}\) directly because we don't know \(\mathrm{k}\) yet
• Strategy: Use the condition \(\mathrm{f(5) = 32}\) to find \(\mathrm{k}\) first, then find \(\mathrm{f(10)}\)

3. SIMPLIFY to find the value of k

• Substitute \(\mathrm{x = 5}\) into \(\mathrm{f(x) = 4x + k(x - 1)}\):
  \(\mathrm{f(5) = 4(5) + k(5 - 1) = 32}\)
  \(\mathrm{20 + k(4) = 32}\)
  \(\mathrm{20 + 4k = 32}\)

• Solve for \(\mathrm{k}\):
  \(\mathrm{4k = 32 - 20 = 12}\)
  \(\mathrm{k = 3}\)

4. SIMPLIFY the function with known k

• Now that \(\mathrm{k = 3}\), substitute back:
  \(\mathrm{f(x) = 4x + 3(x - 1)}\)
  \(\mathrm{f(x) = 4x + 3x - 3}\)
  \(\mathrm{f(x) = 7x - 3}\)

5. SIMPLIFY to find f(10)

• Substitute \(\mathrm{x = 10}\):
  \(\mathrm{f(10) = 7(10) - 3 = 70 - 3 = 67}\)

Answer: 67

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students try to find \(\mathrm{f(10)}\) directly without realizing they need to find \(\mathrm{k}\) first

Students might substitute \(\mathrm{x = 10}\) directly into \(\mathrm{f(x) = 4x + k(x - 1)}\), getting \(\mathrm{f(10) = 40 + 9k}\), then become confused about what to do next since they still have an unknown \(\mathrm{k}\). This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Arithmetic errors when solving for \(\mathrm{k}\)

Students correctly set up \(\mathrm{20 + 4k = 32}\) but make calculation errors like:

• Getting \(\mathrm{4k = 52}\) instead of \(\mathrm{4k = 12}\) (adding instead of subtracting)
• Getting \(\mathrm{k = 4}\) instead of \(\mathrm{k = 3}\) (division error)

With \(\mathrm{k = 4}\), they would get \(\mathrm{f(x) = 8x - 4}\), leading to \(\mathrm{f(10) = 76}\), which doesn't match any typical answer choice and causes confusion.

The Bottom Line:

This problem tests whether students can work backwards from a function value to find unknown parameters before proceeding forward to the final answer. The key insight is recognizing that the given condition \(\mathrm{f(5) = 32}\) is the tool for finding \(\mathrm{k}\), not just extra information.