The function g is defined by \(\mathrm{g(x)} = \sqrt{8\mathrm{x} + 1}\). What is the value of \(\mathrm{g(3)}\)?...

1. TRANSLATE the question

• Given information:
  • Function: \(\mathrm{g(x) = \sqrt{8x + 1}}\)
  • Need to find: \(\mathrm{g(3)}\)
• What this means: Substitute \(\mathrm{x = 3}\) into the function

2. SIMPLIFY by substituting and calculating

• Substitute \(\mathrm{x = 3}\):
  \(\mathrm{g(3) = \sqrt{8(3) + 1}}\)
• Calculate inside the square root first:
  • \(\mathrm{8(3) = 24}\)
  • \(\mathrm{24 + 1 = 25}\)
  • So \(\mathrm{g(3) = \sqrt{25}}\)
• Take the square root:
  \(\mathrm{\sqrt{25} = 5}\)

Answer: C. 5

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students correctly set up \(\mathrm{g(3) = \sqrt{8(3) + 1}}\) and calculate \(\mathrm{8(3) + 1 = 25}\), but then forget to take the square root of 25.

They stop at \(\mathrm{g(3) = 25}\) instead of completing the final step \(\mathrm{g(3) = \sqrt{25} = 5}\).

This leads them to select Choice D (25) instead of the correct answer.

Second Most Common Error:

Poor TRANSLATE reasoning: Students misunderstand function notation and don't know what "find \(\mathrm{g(3)}\)" means, or they substitute incorrectly.

This leads to confusion about where to place the 3 in the expression, causing them to abandon systematic solution and guess among the choices.

The Bottom Line:

This problem tests whether students can execute a complete function evaluation. The key challenge is remembering that when you have a square root expression, you must actually take the square root of your calculated value—don't stop partway through!