The function g is defined by \(\mathrm{g(x) = 3(x + 2)^{-1}}\). What is the value of \(\mathrm{g(7)}\)?1/271/3327

1. TRANSLATE the problem information

• Given: \(\mathrm{g(x) = 3(x + 2)^{-1}}\)
• Find: \(\mathrm{g(7)}\)
• This means we need to substitute \(\mathrm{x = 7}\) into our function definition

2. TRANSLATE by substituting the input value

• Replace every x with 7:

\(\mathrm{g(7) = 3(7 + 2)^{-1}}\)

3. SIMPLIFY the expression inside parentheses first

• Following order of operations, work inside parentheses:

\(\mathrm{g(7) = 3(9)^{-1}}\)

4. INFER that we need to apply the negative exponent rule

• The negative exponent tells us to take the reciprocal
• Remember: \(\mathrm{a^{-1} = \frac{1}{a}}\)
• So \(\mathrm{(9)^{-1} = \frac{1}{9}}\)

5. SIMPLIFY the final multiplication

• \(\mathrm{g(7) = 3 \times \frac{1}{9} = \frac{3}{9} = \frac{1}{3}}\)

Answer: \(\mathrm{\frac{1}{3}}\) (Choice B)

Why Students Usually Falter on This Problem

Most Common Error Path:

Missing conceptual knowledge: Not remembering the negative exponent rule correctly

Students might think that \(\mathrm{(9)^{-1} = -9}\) instead of \(\mathrm{\frac{1}{9}}\). This fundamental misunderstanding of what negative exponents mean leads them to calculate \(\mathrm{g(7) = 3(-9) = -27}\). Since \(\mathrm{-27}\) isn't an answer choice, this leads to confusion and guessing.

Second Most Common Error:

Weak SIMPLIFY execution: Making arithmetic errors during calculation

Students correctly apply the negative exponent rule to get \(\mathrm{g(7) = 3 \times \frac{1}{9}}\), but then make calculation mistakes. They might incorrectly compute \(\mathrm{3 \times \frac{1}{9} = \frac{3}{1} \times \frac{1}{9} = 3}\) instead of \(\mathrm{\frac{3}{9} = \frac{1}{3}}\). This may lead them to select Choice (C) (3).

The Bottom Line:

This problem tests whether students truly understand what negative exponents mean (reciprocals, not negative numbers) and can execute basic function evaluation systematically. The key insight is recognizing that negative exponents create fractions, not negative results.