The function p is defined by \(\mathrm{p(z) = \sqrt{z + 27}}\). What is the value of \(\mathrm{p(9)}\)?

1. TRANSLATE the problem information

• Given information:
  • Function definition: \(\mathrm{p(z) = \sqrt{z + 27}}\)
  • Need to find: \(\mathrm{p(9)}\)
• This tells us we need to substitute 9 for z in the function definition

2. SIMPLIFY through substitution and calculation

• Substitute \(\mathrm{z = 9}\): \(\mathrm{p(9) = \sqrt{9 + 27}}\)
• Perform addition inside the square root: \(\mathrm{9 + 27 = 36}\)
• Calculate the square root: \(\mathrm{\sqrt{36} = 6}\)

Answer: B (6)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may not understand function notation and think \(\mathrm{p(9)}\) means "p times 9" rather than "substitute 9 into the function."

They might attempt to multiply something by 9 or get confused about what the problem is asking for, leading to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors, either in the addition (9 + 27) or in calculating the square root.

Common calculation mistakes include \(\mathrm{9 + 27 = 35}\) (leading them toward answer choice A: 5, since \(\mathrm{\sqrt{35} \approx 5.9}\)) or not recognizing that \(\mathrm{\sqrt{36} = 6}\).

The Bottom Line:

This problem tests whether students understand the fundamental concept of function evaluation - that \(\mathrm{p(9)}\) means "plug 9 into wherever you see the variable z." Students who struggle with function notation will get stuck before they even begin the arithmetic.