The function g is a linear function such that \(\mathrm{g(2) = 5}\) and \(\mathrm{g(4) = 11}\). What is the value...

1. TRANSLATE the problem information

• Given information:
  • g is a linear function
  • \(\mathrm{g(2) = 5}\), which means the point \(\mathrm{(2, 5)}\) is on the line
  • \(\mathrm{g(4) = 11}\), which means the point \(\mathrm{(4, 11)}\) is on the line
  • We need to find \(\mathrm{g(-1)}\)

2. INFER the solution strategy

• Since we need a specific value of a linear function, we must first find the function's equation
• Linear functions have the form \(\mathrm{g(x) = mx + b}\)
• To find m and b, we can use our two known points

3. SIMPLIFY to find the slope

• Using the slope formula with points \(\mathrm{(2, 5)}\) and \(\mathrm{(4, 11)}\):
• \(\mathrm{m = \frac{11 - 5}{4 - 2} = \frac{6}{2} = 3}\)

4. SIMPLIFY to find the y-intercept

• Substitute the slope \(\mathrm{m = 3}\) and point \(\mathrm{(2, 5)}\) into \(\mathrm{g(x) = mx + b}\):
• \(\mathrm{5 = 3(2) + b}\)
• \(\mathrm{5 = 6 + b}\)
• \(\mathrm{b = 5 - 6 = -1}\)

5. SIMPLIFY to find g(-1)

• Now we have \(\mathrm{g(x) = 3x - 1}\)
• \(\mathrm{g(-1) = 3(-1) - 1 = -3 - 1 = -4}\)

Answer: B. -4

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may not recognize that \(\mathrm{g(2) = 5}\) means the point \(\mathrm{(2, 5)}\) lies on the line. Instead, they might try to work directly with the function notation without converting to coordinates, leading to confusion about how to proceed systematically. This leads to abandoning systematic solution and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify the need to find slope and y-intercept but make algebraic errors when solving for b. For example, when solving \(\mathrm{5 = 6 + b}\), they might incorrectly get \(\mathrm{b = 11}\) instead of \(\mathrm{b = -1}\), leading to the wrong function equation. This may lead them to select Choice A (-10) or Choice C (2).

The Bottom Line:

This problem tests whether students understand that function notation represents coordinate points and can systematically build a linear equation from given information. The key insight is recognizing that finding any value of a linear function requires knowing the complete equation first.