The function g is defined by \(\mathrm{g(x) = \begin{cases} 3x - 1, & \text{if } x \leq 5 \\ 2x...

1. TRANSLATE the problem information

• Given information:
  • Function \(\mathrm{g(x)}\) has two pieces: \(\mathrm{3x - 1}\) when \(\mathrm{x \leq 5}\), and \(\mathrm{2x + 4}\) when \(\mathrm{x \gt 5}\)
  • We need to find \(\mathrm{g(7)}\)

2. INFER which piece of the function to use

• Since we're evaluating \(\mathrm{g(7)}\), compare 7 to the boundary value 5
• Because \(\mathrm{7 \gt 5}\), we use the second piece: \(\mathrm{g(x) = 2x + 4}\)

3. SIMPLIFY by substituting \(\mathrm{x = 7}\)

• \(\mathrm{g(7) = 2(7) + 4}\)
• \(\mathrm{g(7) = 14 + 4 = 18}\)

Answer: B (18)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may not carefully check which condition applies to \(\mathrm{x = 7}\), and mistakenly use the first piece of the function (\(\mathrm{3x - 1}\)) instead of the correct second piece (\(\mathrm{2x + 4}\)).

Using the wrong piece: \(\mathrm{g(7) = 3(7) - 1 = 21 - 1 = 20}\)

This may lead them to select Choice C (20)

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify the right piece but make arithmetic errors when calculating \(\mathrm{2(7) + 4}\).

Common calculation mistakes include getting \(\mathrm{2(7) = 16}\) instead of 14, or adding incorrectly at the final step.

This causes them to get stuck or select an incorrect answer choice through computational error.

The Bottom Line:

The key challenge is recognizing that piecewise functions require you to first determine which 'piece' applies before doing any calculations. Students who jump straight into computation without checking the conditions will use the wrong formula entirely.