Functions f and g are defined by \(\mathrm{f(x) = 2x + k}\) and \(\mathrm{g(x) = x^2 - 3x}\). In the...

1. TRANSLATE the composite function notation

• Given information:
  • \(\mathrm{f(x) = 2x + k}\) (where k is unknown)
  • \(\mathrm{g(x) = x^2 - 3x}\)
  • \(\mathrm{g(f(x)) = 4x^2 + 18x + 18}\)

• What \(\mathrm{g(f(x))}\) means: Substitute the entire expression \(\mathrm{f(x)}\) wherever you see x in \(\mathrm{g(x)}\)

2. TRANSLATE g(f(x)) into substitution form

• \(\mathrm{g(f(x)) = g(2x + k)}\)
• Now substitute \(\mathrm{(2x + k)}\) for every x in \(\mathrm{g(x)}\):
• \(\mathrm{g(2x + k) = (2x + k)^2 - 3(2x + k)}\)

3. SIMPLIFY by expanding the algebraic expression

• Expand \(\mathrm{(2x + k)^2}\):
  \(\mathrm{(2x + k)^2 = (2x)(2x) + 2(2x)(k) + k^2}\)
  \(\mathrm{= 4x^2 + 4kx + k^2}\)

• Expand \(\mathrm{-3(2x + k)}\):
  \(\mathrm{-3(2x + k) = -6x - 3k}\)

• Combine everything:
  \(\mathrm{g(f(x)) = 4x^2 + 4kx + k^2 - 6x - 3k}\)
  \(\mathrm{g(f(x)) = 4x^2 + (4k - 6)x + (k^2 - 3k)}\)

4. INFER the solution strategy from polynomial equality

• We know: \(\mathrm{g(f(x)) = 4x^2 + 18x + 18}\)
• We found: \(\mathrm{g(f(x)) = 4x^2 + (4k - 6)x + (k^2 - 3k)}\)
• For these polynomials to be equal, their coefficients must match term by term

5. INFER the equations by comparing coefficients

• \(\mathrm{x^2}\) coefficients: \(\mathrm{4 = 4}\) ✓ (this checks out)
• x coefficients: \(\mathrm{4k - 6 = 18}\)
• constant terms: \(\mathrm{k^2 - 3k = 18}\)

6. SIMPLIFY to solve for k

• From the x coefficient equation:
  \(\mathrm{4k - 6 = 18}\)
  \(\mathrm{4k = 24}\)
  \(\mathrm{k = 6}\)

7. APPLY CONSTRAINTS by verifying the solution

• Check using the constant term equation:
  \(\mathrm{k^2 - 3k = 6^2 - 3(6)}\)
  \(\mathrm{= 36 - 18}\)
  \(\mathrm{= 18}\) ✓
• Since \(\mathrm{18 = 18}\), our solution is correct

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Not understanding that \(\mathrm{g(f(x))}\) requires substituting the entire expression \(\mathrm{f(x) = 2x + k}\) into \(\mathrm{g(x)}\)

Students might try to work with \(\mathrm{g(f(x))}\) as separate functions or attempt to "multiply" g and f together. Some might substitute \(\mathrm{x = 2x + k}\) incorrectly, leading to confused algebraic manipulations.

This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Making algebraic errors when expanding \(\mathrm{(2x + k)^2}\)

Students commonly expand this as \(\mathrm{4x^2 + k^2}\) (forgetting the middle term \(\mathrm{4kx}\)) or make sign errors when distributing \(\mathrm{-3(2x + k)}\). These errors create incorrect coefficient equations.

This may lead them to select Choice A (-3) or Choice B (3) depending on the specific algebraic mistake.

The Bottom Line:

This problem tests whether students can translate composite function notation into concrete substitution and then execute multi-step algebraic manipulation accurately. The conceptual hurdle is understanding what \(\mathrm{g(f(x))}\) actually means, while the execution challenge lies in errorless polynomial expansion.