Question:A linear function g satisfies \(\mathrm{g(2) - g(-3) = 10}\) and \(\mathrm{g(0) = 5}\). Which of the following defines \(\mathrm{g(x)}\)?\(\m...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{g(2) - g(-3) = 10}\)
  • \(\mathrm{g(0) = 5}\)
  • g is a linear function
• We need to find which answer choice correctly defines g(x)

2. INFER the solution approach

• Since g is linear, it has the form \(\mathrm{g(x) = mx + b}\)
• The condition \(\mathrm{g(0) = 5}\) will give us the y-intercept b directly
• The difference condition \(\mathrm{g(2) - g(-3) = 10}\) will help us find the slope m

3. SIMPLIFY to find the y-intercept

• From \(\mathrm{g(0) = 5}\):
• \(\mathrm{g(0) = m(0) + b = 0 + b = b}\)
• Therefore: \(\mathrm{b = 5}\)

4. TRANSLATE the difference condition into algebra

• \(\mathrm{g(2) = m(2) + b = 2m + 5}\)
• \(\mathrm{g(-3) = m(-3) + b = -3m + 5}\)
• So: \(\mathrm{g(2) - g(-3) = (2m + 5) - (-3m + 5)}\)

5. SIMPLIFY the algebraic expression

• \(\mathrm{g(2) - g(-3) = (2m + 5) - (-3m + 5)}\)
• \(\mathrm{= 2m + 5 + 3m - 5}\)
• \(\mathrm{= 5m}\)

6. INFER the value of the slope

• Since \(\mathrm{g(2) - g(-3) = 10}\) and we found \(\mathrm{g(2) - g(-3) = 5m}\):
• \(\mathrm{5m = 10}\)
• \(\mathrm{m = 2}\)

7. Combine results

• We found \(\mathrm{m = 2}\) and \(\mathrm{b = 5}\)
• Therefore: \(\mathrm{g(x) = 2x + 5}\)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make sign errors when expanding \(\mathrm{g(2) - g(-3) = (2m + 5) - (-3m + 5)}\)

They might incorrectly get: \(\mathrm{(2m + 5) - (-3m + 5) = 2m + 5 - 3m + 5 = -m + 10}\)

This leads to \(\mathrm{-m = 10}\), so \(\mathrm{m = -10}\), giving them \(\mathrm{g(x) = -10x + 5}\), which doesn't match any answer choice. This causes confusion and guessing.

Second Most Common Error:

Poor TRANSLATE reasoning: Students confuse which values to substitute where, potentially mixing up g(2) and g(-3) or incorrectly setting up the difference equation.

This leads to incorrect slope calculations, possibly selecting Choice A (\(\mathrm{g(x) = -2x + 5}\)) if they get the wrong sign for the slope.

The Bottom Line:

This problem tests careful algebraic manipulation of the difference between function values. The key insight is recognizing that for linear functions, differences in function values relate directly to the slope through the difference in input values.