The function f is defined by \(\mathrm{f(x) = 5x^2}\). What is the value of \(\mathrm{f(8)}\)?

1. TRANSLATE the problem information

• Given information:
  • Function definition: \(\mathrm{f(x) = 5x^2}\)
  • Need to find: \(\mathrm{f(8)}\)
• This means we need to substitute 8 for x in the function

2. SIMPLIFY through substitution and calculation

• Substitute \(\mathrm{x = 8}\) into \(\mathrm{f(x) = 5x^2}\):
  \(\mathrm{f(8) = 5(8)^2}\)
• Calculate the exponent first: \(\mathrm{8^2 = 64}\)
  \(\mathrm{f(8) = 5(64)}\)
• Multiply: \(\mathrm{f(8) = 320}\)

Answer: D. 320

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students may calculate \(\mathrm{8^2}\) incorrectly, thinking \(\mathrm{8^2 = 16}\) instead of 64.

This leads them to compute \(\mathrm{f(8) = 5(16) = 80}\), causing them to select Choice C (80).

Second Most Common Error:

Poor order of operations in SIMPLIFY: Students might multiply \(\mathrm{5 \times 8 = 40}\) first, then square the result: \(\mathrm{(40)^2 = 1600}\), getting a result not among the choices.

This leads to confusion and guessing among the available options.

The Bottom Line:

This problem tests basic function evaluation skills, but the key challenge lies in careful arithmetic execution, particularly remembering that \(\mathrm{8^2 = 64}\), not 16.