If g is the function defined by \(\mathrm{g(x) = \frac{3x + 2}{4}}\), and \(\mathrm{g(a) = 5}\), what is the value...

1. TRANSLATE the problem information

• Given information:
  • Function: \(\mathrm{g(x) = \frac{3x + 2}{4}}\)
  • Condition: \(\mathrm{g(a) = 5}\)
  • Need to find: the value of a

• This tells us we need to substitute a into the function and set it equal to 5

2. TRANSLATE the condition into an equation

• Since \(\mathrm{g(a) = 5}\), we substitute a for x in the function:
  \(\mathrm{g(a) = \frac{3a + 2}{4} = 5}\)

3. SIMPLIFY by solving the linear equation

• Start with: \(\mathrm{\frac{3a + 2}{4} = 5}\)
• Multiply both sides by 4: \(\mathrm{3a + 2 = 20}\)
• Subtract 2 from both sides: \(\mathrm{3a = 18}\)
• Divide both sides by 3: \(\mathrm{a = 6}\)

4. Verify the answer

• Check: \(\mathrm{g(6) = \frac{3(6) + 2}{4} = \frac{18 + 2}{4} = \frac{20}{4} = 5}\) ✓

Answer: D. 6

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students might incorrectly set up the initial equation, perhaps writing \(\mathrm{g(x) = 5}\) instead of recognizing they need to substitute the unknown value a into the function. This fundamental misunderstanding prevents them from creating the correct equation \(\mathrm{\frac{3a + 2}{4} = 5}\), leading to confusion and random answer selection.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{\frac{3a + 2}{4} = 5}\) but make arithmetic errors when solving. Common mistakes include forgetting to multiply both sides by 4, sign errors when subtracting 2, or division mistakes in the final step. These calculation errors often lead them to select Choice A (2) or Choice B \(\mathrm{\frac{17}{4}}\).

The Bottom Line:

This problem tests the fundamental connection between function notation and equation solving. Success requires accurately translating the function evaluation condition into an algebraic equation, then executing clean algebraic manipulation without arithmetic errors.