The function g satisfies \(\mathrm{g(x + 2) = 3x - 8}\) for all real numbers x. What is the value...

1. TRANSLATE the problem information

• Given: \(\mathrm{g(x + 2) = 3x - 8}\) for all real x
• Find: \(\mathrm{g(9)}\)
• What this tells us: We have a function where the input is expressed as \(\mathrm{(x + 2)}\), not just \(\mathrm{x}\)

2. INFER the solution strategy

• To find \(\mathrm{g(9)}\), we need to use the given relationship \(\mathrm{g(x + 2) = 3x - 8}\)
• Key insight: We need to determine what value of x makes \(\mathrm{x + 2 = 9}\)
• This will allow us to substitute that x-value into \(\mathrm{3x - 8}\)

3. Set up the equation to find the appropriate x-value

• We need: \(\mathrm{x + 2 = 9}\)
• Solving: \(\mathrm{x = 7}\)

4. SIMPLIFY by substituting into the given expression

• Now we know that \(\mathrm{g(9) = g(7 + 2)}\)
  \(\mathrm{= 3(7) - 8}\)
• Calculate:
  \(\mathrm{3(7) - 8}\)
  \(\mathrm{= 21 - 8}\)
  \(\mathrm{= 13}\)

Answer: A. 13

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students attempt to substitute \(\mathrm{x = 9}\) directly into \(\mathrm{3x - 8}\), thinking:
\(\mathrm{g(9) = 3(9) - 8}\)
\(\mathrm{= 27 - 8}\)
\(\mathrm{= 19}\)

They miss the crucial insight that the function is defined in terms of \(\mathrm{(x + 2)}\), not \(\mathrm{x}\). They don't realize they need to work backwards to find which x-value produces \(\mathrm{x + 2 = 9}\).

This may lead them to select Choice B (19).

The Bottom Line:

This problem tests whether students understand that functional relationships can have transformed inputs. The key breakthrough is recognizing that you must match the input structure of the given function, not just substitute the desired output value directly.