The function f is defined by \(\mathrm{f(x) = x^3 + 15}\). What is the value of \(\mathrm{f(2)}\)?

1. TRANSLATE the problem information

• Given information:
  • Function: \(\mathrm{f(x) = x^3 + 15}\)
  • Need to find: \(\mathrm{f(2)}\)

• What \(\mathrm{f(2)}\) means: Substitute \(\mathrm{x = 2}\) into the function \(\mathrm{f(x)}\)

2. SIMPLIFY by substituting and calculating

• Replace every x in \(\mathrm{f(x) = x^3 + 15}\) with 2:
  \(\mathrm{f(2) = (2)^3 + 15}\)

• Calculate the exponent first:
  \(\mathrm{(2)^3 = 2 \times 2 \times 2 = 8}\)

• Complete the addition:
  \(\mathrm{f(2) = 8 + 15 = 23}\)

Answer: C. 23

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may not clearly understand what \(\mathrm{f(2)}\) means or how function notation works. They might think \(\mathrm{f(2)}\) means "f times 2" or get confused about the substitution process.

This leads to confusion and guessing rather than systematic substitution.

Second Most Common Error:

Poor SIMPLIFY execution: Students might misread the function notation and interpret \(\mathrm{x^3}\) as \(\mathrm{x(3)}\) instead of x cubed. This would give them \(\mathrm{f(2) = 2(3) + 15 = 6 + 15 = 21}\).

This may lead them to select Choice B (21).

The Bottom Line:

This problem tests whether students understand the fundamental concept of function evaluation - that \(\mathrm{f(2)}\) means substitute 2 for every instance of x in the function. The arithmetic is straightforward once students correctly set up the substitution.