The function g is defined by \(\mathrm{g(x) = \frac{1}{3}x^2}\). What is the value of \(\mathrm{g(9)}\)?381927

1. TRANSLATE the problem information

• Given: \(\mathrm{g(x) = \frac{1}{3}x^2}\)
• Find: \(\mathrm{g(9)}\)
• What this means: Replace x with 9 in the function definition

2. TRANSLATE the substitution

• \(\mathrm{g(9) = \frac{1}{3}(9)^2}\)
• We now have a numerical expression to evaluate

3. SIMPLIFY using order of operations

• Calculate the exponent first: \(\mathrm{9^2 = 81}\)
• Expression becomes: \(\mathrm{g(9) = \frac{1}{3}(81)}\)

4. SIMPLIFY the final multiplication

• \(\mathrm{\frac{1}{3}(81) = 81 \div 3 = 27}\)

Answer: D. 27

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Not following correct order of operations

Students calculate \(\mathrm{\frac{1}{3} \times 9}\) first, getting 3, then square this result: \(\mathrm{3^2 = 9}\).

This reasoning: "I'll multiply 1/3 by 9 to get 3, then square it" violates order of operations since exponents should be calculated before multiplication.

This may lead them to select Choice C (9).

Second Most Common Error:

Incomplete SIMPLIFY execution: Forgetting the coefficient entirely

Students focus only on the x² part, calculating \(\mathrm{9^2 = 81}\), but forget to multiply by the \(\mathrm{\frac{1}{3}}\) coefficient.

This may lead them to select Choice B (81).

The Bottom Line:

Order of operations is crucial in function evaluation - the exponent must be calculated before any multiplication with coefficients, even when fractions are involved.