Let g be a function defined for all real numbers such that \(\mathrm{g(x + 1) = x^3 + 15}\). What...

1. TRANSLATE the problem information

• Given information: \(\mathrm{g(x + 1) = x^3 + 15}\)
• Need to find: \(\mathrm{g(3)}\)
• This means we need to find the output when the input to function g is 3

2. INFER the solution strategy

• The function is defined in terms of \(\mathrm{g(x + 1)}\), not \(\mathrm{g(x)}\)
• To find \(\mathrm{g(3)}\), we need to figure out what value of x makes \(\mathrm{x + 1 = 3}\)
• Key insight: Set up the equation \(\mathrm{x + 1 = 3}\)

3. SIMPLIFY to find the x-value

• Solve \(\mathrm{x + 1 = 3}\)
• Therefore \(\mathrm{x = 2}\)

4. SIMPLIFY to evaluate the function

• Now substitute \(\mathrm{x = 2}\) into the original function:
• \(\mathrm{g(3) = g(2 + 1) = 2^3 + 15}\)
• Calculate: \(\mathrm{2^3 = 8}\)
• Final calculation: \(\mathrm{8 + 15 = 23}\)

Answer: C. 23

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the substitution strategy and instead try to plug \(\mathrm{x = 3}\) directly into \(\mathrm{x^3 + 15}\).

They think: "I need \(\mathrm{g(3)}\), so I'll use \(\mathrm{x = 3}\)" and calculate \(\mathrm{3^3 + 15 = 27 + 15 = 42}\). When they don't see 42 among the choices, they might just pick \(\mathrm{3^3 = 27}\).

This may lead them to select Choice (E) (27).

Second Most Common Error:

Poor TRANSLATE reasoning: Students misunderstand what the functional notation means and get confused about which variable represents the input.

They might set \(\mathrm{g(3) = 3^3 + 15}\) directly, thinking that since we want \(\mathrm{g(3)}\), we substitute 3 for x everywhere.

This leads to confusion and potentially guessing among the available choices.

The Bottom Line:

The key challenge is recognizing that when a function is defined as \(\mathrm{g(x + 1)}\), finding \(\mathrm{g(3)}\) requires working backwards to determine what x-value makes \(\mathrm{x + 1 = 3}\). This substitution insight is what separates successful students from those who get stuck.