The function g is defined by \(\mathrm{g(x) = 18(1/3)^x}\). What is \(\mathrm{g(-1) + g(0) + g(1)}\)?30607278

1. TRANSLATE the problem information

• Given information:
  • Function: \(\mathrm{g(x) = 18(1/3)^x}\)
  • Need to find: \(\mathrm{g(-1) + g(0) + g(1)}\)
• This means we need to substitute \(\mathrm{x = -1}\), \(\mathrm{x = 0}\), and \(\mathrm{x = 1}\) into our function, then add the results

2. INFER the approach needed

• We'll evaluate each function value separately using exponent rules
• Key insight: We need to handle negative exponents and zero exponents correctly
• Then we'll add all three results

3. SIMPLIFY each function evaluation

For g(-1):

\(\mathrm{g(-1) = 18(1/3)^{-1}}\)

Using the rule that \(\mathrm{a^{-n} = 1/a^n}\):

\(\mathrm{(1/3)^{-1} = 1/(1/3) = 3}\)

So \(\mathrm{g(-1) = 18 × 3 = 54}\)

For g(0):

\(\mathrm{g(0) = 18(1/3)^0}\)

Using the rule that any non-zero number to the 0 power equals 1:

\(\mathrm{(1/3)^0 = 1}\)

So \(\mathrm{g(0) = 18 × 1 = 18}\)

For g(1):

\(\mathrm{g(1) = 18(1/3)^1 = 18 × (1/3) = 6}\)

4. SIMPLIFY the final calculation

\(\mathrm{g(-1) + g(0) + g(1) = 54 + 18 + 6 = 78}\)

Answer: D (78)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students mishandle negative exponents, thinking that \(\mathrm{(1/3)^{-1} = -1/3}\) instead of applying the rule correctly to get 3.

This leads them to calculate \(\mathrm{g(-1) = 18 × (-1/3) = -6}\), giving a final sum of \(\mathrm{-6 + 18 + 6 = 18}\). However, since 18 isn't among the answer choices, this typically leads to confusion and guessing.

Second Most Common Error:

Missing conceptual knowledge: Students forget that any non-zero number raised to the 0 power equals 1, so they might think \(\mathrm{(1/3)^0 = 0}\).

This would give them \(\mathrm{g(0) = 18 × 0 = 0}\), leading to a final sum of \(\mathrm{54 + 0 + 6 = 60}\). This may lead them to select Choice B (60).

The Bottom Line:

This problem tests whether students can correctly apply exponent rules, especially the counterintuitive negative exponent rule where \(\mathrm{(1/3)^{-1} = 3}\), not a negative number. The key insight is recognizing that negative exponents flip fractions and zero exponents always give 1.