The function g is defined by \(\mathrm{g(x) = x + 6}\), if x gt 0; \(\mathrm{g(x) = x - 6}\),...

1. TRANSLATE the function definition

• Given: \(\mathrm{g(x) = \{x + 6,\ if\ x \gt 0;\ x - 6,\ if\ x \leq 0\}}\)
• This tells us we have two different rules:
  • When x is positive: use \(\mathrm{g(x) = x + 6}\)
  • When x is zero or negative: use \(\mathrm{g(x) = x - 6}\)

2. INFER which piece applies to our input

• We need \(\mathrm{g(4)}\)
• Since \(\mathrm{4 \gt 0}\), we use the first piece: \(\mathrm{g(x) = x + 6}\)

3. SIMPLIFY by substituting and calculating

• \(\mathrm{g(4) = 4 + 6 = 10}\)

Answer: 10

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students misapply the conditions and use the wrong piece of the function.

They might think "4 is just a number" without carefully checking which condition it satisfies, or they might confuse the inequality signs. This leads them to use \(\mathrm{g(x) = x - 6}\) instead of \(\mathrm{g(x) = x + 6}\), giving \(\mathrm{g(4) = 4 - 6 = -2}\). This causes them to get stuck or guess incorrectly.

Second Most Common Error:

Poor TRANSLATE reasoning: Students don't fully understand piecewise function notation.

They might try to use both pieces simultaneously or get confused about when each rule applies. This leads to confusion about the basic setup of the problem, causing them to abandon systematic solution and guess.

The Bottom Line:

Piecewise functions require careful attention to conditions. The key insight is that you must first determine which "piece" or rule applies before you can evaluate the function - it's not about doing complex math, it's about logical reasoning.