The function f is defined by \(\mathrm{f(x) = 8x^3 + 4}\). What is the value of \(\mathrm{f(2)}\)?

1. TRANSLATE the problem information

• Given information:
  • Function f is defined by \(\mathrm{f(x) = 8x^3 + 4}\)
  • Need to find \(\mathrm{f(2)}\)
• What this tells us: We need to substitute \(\mathrm{x = 2}\) into the function

2. INFER the approach

• To find \(\mathrm{f(2)}\), substitute 2 for every x in the function
• Replace x with 2 in \(\mathrm{f(x) = 8x^3 + 4}\)

3. SIMPLIFY through the calculation

• \(\mathrm{f(2) = 8(2)^3 + 4}\)
• First evaluate the exponent: \(\mathrm{2^3 = 2 \times 2 \times 2 = 8}\)
• \(\mathrm{f(2) = 8(8) + 4}\)
• Next multiply: \(\mathrm{8 \times 8 = 64}\)
• \(\mathrm{f(2) = 64 + 4}\)
• Finally add: \(\mathrm{f(2) = 68}\)

Answer: 68

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Student doesn't understand function notation and what \(\mathrm{f(2)}\) means

Some students treat \(\mathrm{f(2)}\) as multiplication, thinking it means \(\mathrm{f \times 2}\), rather than understanding it means "substitute 2 for x in the function." Others might not know how to proceed with substitution. This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Student makes arithmetic errors with order of operations

Student correctly understands the substitution but makes calculation mistakes. Common errors include:

• Calculating \(\mathrm{2^3 = 6}\) instead of 8
• Forgetting to multiply \(\mathrm{8 \times 8}\), writing \(\mathrm{f(2) = 8 + 8 + 4 = 20}\)
• Adding before multiplying, getting \(\mathrm{f(2) = 8 \times (8 + 4) = 96}\)

These arithmetic mistakes lead to various incorrect answers.

The Bottom Line:

This problem tests whether students understand function notation and can execute multi-step arithmetic correctly. Success requires both conceptual understanding of functions and careful attention to order of operations.