For the linear function g, the table shows three values of x and their corresponding values of \(\mathrm{g(x)}\). If \(\mathrm{j(x)...

1. INFER the solution strategy

• Given information: Table of x and g(x) values, and transformation \(\mathrm{j(x) = -g(x) + 8}\)
• What this tells us: We need to find g(x) first, then apply the transformation
• Key insight: Since we have three points and know g is linear, we can determine g(x) completely, then transform it to get j(x)

2. SIMPLIFY to find the slope of g(x)

• Pick any two points from the table: \((-4/3, 3)\) and \((-5/3, 2)\)
• Apply slope formula: \(\mathrm{m = \frac{2-3}{-5/3-(-4/3)} = \frac{-1}{-1/3} = 3}\)

3. INFER the complete equation of g(x)

• Since slope = 3, we have \(\mathrm{g(x) = 3x + b}\)
• Use any point to find b. Using (-1, 4):
  \(\mathrm{4 = 3(-1) + b}\)
  \(\mathrm{4 = -3 + b}\)
  \(\mathrm{b = 7}\)
• Therefore: \(\mathrm{g(x) = 3x + 7}\)

4. SIMPLIFY by verifying with remaining point

• Check \(\mathrm{g(-4/3) = 3(-4/3) + 7 = -4 + 7 = 3}\) ✓
• This confirms our equation is correct

5. SIMPLIFY the transformation to get j(x)

• Apply \(\mathrm{j(x) = -g(x) + 8}\):
  \(\mathrm{j(x) = -(3x + 7) + 8}\)
  \(\mathrm{j(x) = -3x - 7 + 8}\)
  \(\mathrm{j(x) = -3x + 1}\)

Answer: B) \(\mathrm{j(x) = -3x + 1}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students correctly find \(\mathrm{g(x) = 3x + 7}\) but make sign errors when applying the transformation \(\mathrm{j(x) = -g(x) + 8}\). They might distribute the negative incorrectly, getting \(\mathrm{j(x) = -3x + 7 + 8 = -3x + 15}\), or forget to distribute it at all, getting \(\mathrm{j(x) = 3x - 7 + 8 = 3x + 1}\).

This may lead them to select Choice E (\(\mathrm{-3x + 15}\)) or Choice C (\(\mathrm{3x + 1}\)).

Second Most Common Error:

Poor INFER reasoning: Students attempt to work backwards from the answer choices instead of systematically finding g(x) first. This leads to guessing or trying to match patterns without understanding the underlying relationship.

This leads to confusion and random answer selection among the choices.

The Bottom Line:

This problem tests both systematic linear function analysis and careful algebraic manipulation. Success requires methodically finding the original function before applying transformations, combined with precise attention to signs during algebraic simplification.