The function g is defined by \(\mathrm{g(y) = 2y^{1/2} + 9}\). What is the value of \(\mathrm{g(36)}\)?

1. TRANSLATE the problem information

• Given: Function \(\mathrm{g(y) = 2y^{1/2} + 9}\)
• Find: \(\mathrm{g(36)}\)
• This means: substitute 36 for y in the function

2. TRANSLATE the mathematical notation

• The expression \(\mathrm{y^{1/2}}\) means the square root of y
• So our function becomes: \(\mathrm{g(y) = 2\sqrt{y} + 9}\)

3. SIMPLIFY by substitution

• Replace y with 36: \(\mathrm{g(36) = 2\sqrt{36} + 9}\)
• Evaluate the square root: \(\mathrm{\sqrt{36} = 6}\)
• Continue: \(\mathrm{g(36) = 2(6) + 9}\)

4. SIMPLIFY using order of operations

• Multiply first: \(\mathrm{2 × 6 = 12}\)
• Then add: \(\mathrm{12 + 9 = 21}\)

Answer: C. 21

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Misinterpreting the exponent notation \(\mathrm{y^{1/2}}\)

Many students see \(\mathrm{y^{1/2}}\) and think "half of y" rather than "square root of y." With this misconception, they calculate \(\mathrm{(36)^{1/2} = 36 ÷ 2 = 18}\), then continue: \(\mathrm{g(36) = 2(18) + 9 = 36 + 9 = 45}\).

This leads them to select Choice D (45).

Second Most Common Error:

Incomplete SIMPLIFY execution: Forgetting to multiply by the coefficient 2

Some students correctly recognize that \(\mathrm{\sqrt{36} = 6}\), but then forget to multiply by 2. They calculate \(\mathrm{g(36) = 6 + 9 = 15}\) instead of \(\mathrm{2(6) + 9 = 21}\).

This may lead them to select Choice A (15).

The Bottom Line:

This problem tests whether students truly understand exponent notation beyond just memorizing that \(\mathrm{x^2}\) means "x squared." The fractional exponent 1/2 requires recognizing the connection to square roots, which is a crucial algebraic concept.