Question:\(\mathrm{f(x) = x^2 + ax + 1}\)\(\mathrm{h(x) = 2x - 3}\)Functions f and h are given, and in function f,...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{f(x) = x^2 + ax + 1}\) (where a is unknown)
  • \(\mathrm{h(x) = 2x - 3}\)
  • \(\mathrm{f(h(x)) = 4x^2 - 10x + 7}\)
• We need to find the value of a

2. INFER the solution strategy

• Since we know what \(\mathrm{f(h(x))}\) equals, we can find it by substituting \(\mathrm{h(x)}\) into \(\mathrm{f(x)}\)
• Then we'll compare our result with the given expression \(\mathrm{4x^2 - 10x + 7}\)
• The key insight: two polynomials are equal if their corresponding coefficients match

3. SIMPLIFY by substituting h(x) into f(x)

• \(\mathrm{f(h(x)) = f(2x - 3) = (2x - 3)^2 + a(2x - 3) + 1}\)
• First, expand \(\mathrm{(2x - 3)^2}\):
  \(\mathrm{(2x - 3)^2 = 4x^2 - 12x + 9}\)

4. SIMPLIFY by combining all terms

• \(\mathrm{f(h(x)) = 4x^2 - 12x + 9 + a(2x - 3) + 1}\)
• \(\mathrm{f(h(x)) = 4x^2 - 12x + 9 + 2ax - 3a + 1}\)
• \(\mathrm{f(h(x)) = 4x^2 + (2a - 12)x + (10 - 3a)}\)

5. INFER by comparing coefficients

• Our expression: \(\mathrm{4x^2 + (2a - 12)x + (10 - 3a)}\)
• Given expression: \(\mathrm{4x^2 - 10x + 7}\)
• For these to be equal, coefficients of like terms must match:
  • \(\mathrm{x^2}\) terms: \(\mathrm{4 = 4}\) ✓
  • \(\mathrm{x}\) terms: \(\mathrm{2a - 12 = -10}\)
  • Constant terms: \(\mathrm{10 - 3a = 7}\)

6. SIMPLIFY by solving for a

• From the x coefficient: \(\mathrm{2a - 12 = -10}\)
• \(\mathrm{2a = 2}\)
• \(\mathrm{a = 1}\)

7. Verify with the constant term

• \(\mathrm{10 - 3a = 10 - 3(1) = 7}\) ✓

Answer: D (1)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize they need to substitute the entire expression \(\mathrm{h(x) = 2x - 3}\) into \(\mathrm{f(x)}\). Instead, they might try to work backwards from \(\mathrm{f(h(x)) = 4x^2 - 10x + 7}\) or substitute specific values.

Without the systematic substitution approach, students get stuck trying various unsuccessful strategies and often resort to guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly substitute \(\mathrm{h(x)}\) into \(\mathrm{f(x)}\) but make algebraic errors when expanding \(\mathrm{(2x - 3)^2}\) or when distributing \(\mathrm{a(2x - 3)}\). A common mistake is getting \(\mathrm{(2x - 3)^2 = 4x^2 - 6x + 9}\) instead of \(\mathrm{4x^2 - 12x + 9}\).

This algebraic error leads to incorrect coefficient comparison and wrong values for a. This may lead them to select Choice B (-2) or get confused and guess.

The Bottom Line:

This problem tests whether students understand function composition conceptually (not just mechanically) and can execute multi-step polynomial algebra accurately. The key insight is recognizing that polynomial equality means coefficient equality.