Question:The function g is defined by \(\mathrm{g(x) = \frac{3x}{4} + 18}\). What value of x satisfies the equation \(\mathrm{g(x) =...

1. TRANSLATE the problem information

• Given information:
  • Function: \(\mathrm{g(x) = \frac{3x}{4} + 18}\)
  • Need to find: value of x when \(\mathrm{g(x) = 30}\)

• This tells us we need to substitute the function definition into the equation \(\mathrm{g(x) = 30}\)

2. TRANSLATE to set up the equation

• Since \(\mathrm{g(x) = \frac{3x}{4} + 18}\) and we want \(\mathrm{g(x) = 30}\):
• Replace g(x) with its definition: \(\mathrm{\frac{3x}{4} + 18 = 30}\)

3. SIMPLIFY to isolate the x-term

• Subtract 18 from both sides: \(\mathrm{\frac{3x}{4} = 12}\)
• This removes the constant term and leaves us with just the x-term on one side

4. SIMPLIFY to solve for x

• Multiply both sides by 4/3 (the reciprocal of 3/4):
• \(\mathrm{x = 12 \times \frac{4}{3}}\)
• \(\mathrm{x = \frac{48}{3}}\)
• \(\mathrm{x = 16}\)

5. Verify the answer

• Check: \(\mathrm{g(16) = \frac{3(16)}{4} + 18}\)
• \(\mathrm{g(16) = 12 + 18}\)
• \(\mathrm{g(16) = 30}\) ✓

Answer: 16

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students may not correctly substitute the function definition into the equation \(\mathrm{g(x) = 30}\). They might try to work with \(\mathrm{g(x) = 30}\) directly without recognizing that they need to replace g(x) with \(\mathrm{\frac{3x}{4} + 18}\).

This leads to confusion about how to proceed and often results in guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors when working with fractions, particularly when multiplying \(\mathrm{12 \times \frac{4}{3}}\). They might incorrectly calculate this as \(\mathrm{12 \times 4 \div 3 = 48 \div 3 = 16}\), or make errors in the fraction arithmetic, leading to answers like 9 or 36.

This causes them to get an incorrect final answer even though their approach was correct.

The Bottom Line:

This problem tests whether students can bridge the gap between function notation and algebraic manipulation. The key insight is recognizing that solving \(\mathrm{g(x) = 30}\) requires substituting the function's definition, then applying systematic algebraic steps to isolate x.