\(\mathrm{g(x) = 8(2^{(x-1)})}\)Which table gives three values of x and their corresponding values of g(x) for function g?x012\(\mathrm{g(x)}\)1/16116...

1. TRANSLATE the problem information

• Given function: \(\mathrm{g(x) = 8(2^{x-1})}\)
• Need to find: Which table shows correct \(\mathrm{g(x)}\) values for \(\mathrm{x = 0, 1, 2}\)
• What this tells us: We must evaluate the function at each x-value and compare with the tables

2. SIMPLIFY by evaluating g(0)

• \(\mathrm{g(0) = 8(2^{0-1}) = 8(2^{-1})}\)
• Since \(\mathrm{2^{-1} = \frac{1}{2}}\): \(\mathrm{g(0) = 8(\frac{1}{2}) = 4}\)
• This immediately eliminates choices A and D (which show 1/16 and 8 respectively)

3. SIMPLIFY by evaluating g(1)

• \(\mathrm{g(1) = 8(2^{1-1}) = 8(2^0)}\)
• Since \(\mathrm{2^0 = 1}\): \(\mathrm{g(1) = 8(1) = 8}\)
• This eliminates choice B (which shows \(\mathrm{g(1) = 4}\))

4. SIMPLIFY by evaluating g(2) to confirm

• \(\mathrm{g(2) = 8(2^{2-1}) = 8(2^1) = 8(2) = 16}\)
• This matches choice C

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill with negative exponents: Students often struggle with \(\mathrm{2^{-1}}\), either forgetting that negative exponents mean "reciprocal" or making calculation errors like thinking \(\mathrm{2^{-1} = -2}\) or \(\mathrm{2^{-1} = -\frac{1}{2}}\).

When evaluating \(\mathrm{g(0) = 8(2^{-1})}\), they might calculate:

• \(\mathrm{8(-2) = -16}\), or
• \(\mathrm{8(-\frac{1}{2}) = -4}\), or
• Get confused and guess

This leads to confusion and guessing since none of the answer choices contain negative values.

Second Most Common Error:

Poor order of operations in SIMPLIFY: Students might multiply 8 × 2 first, then apply the exponent: \(\mathrm{8 \times 2^{x-1}}\) becomes \(\mathrm{(8 \times 2)^{x-1} = 16^{x-1}}\).

For \(\mathrm{x = 0}\): \(\mathrm{16^{-1} = \frac{1}{16}}\), which matches choice A
For \(\mathrm{x = 1}\): \(\mathrm{16^0 = 1}\), which matches choice A
For \(\mathrm{x = 2}\): \(\mathrm{16^1 = 16}\), which matches choice A

This systematic error may lead them to select Choice A.

The Bottom Line:

This problem tests whether students can correctly handle negative exponents and follow proper order of operations in exponential expressions. The key insight is recognizing that exponents are calculated before multiplication, and that \(\mathrm{2^{-1}}\) equals \(\mathrm{\frac{1}{2}}\), not a negative number.