Question

what is the decay factor of the exponential function represented by the table?

\

$$\begin{array}{|c|c|} \\hline x & f(x) \\\\ \\hline -1 & 18 \\\\ \\hline 0 & 6 \\\\ \\hline 1 & 2 \\\\ \\hline 2 & \\frac{2}{3} \\\\ \\hline \\end{array}$$

• \\(\frac{1}{3}\\)
• \\(\frac{2}{3}\\)
• \\(2\\)
• \\(6\\)

Explanation:

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<reasoning>

Identify the exponential function form

\[
f(x) = a \cdot b^x
\]

Calculate the ratio of consecutive terms

\[
b = \frac{f(x+1)}{f(x)} = \frac{f(0)}{f(-1)} = \frac{6}{18} = \frac{1}{3}
\]

Verify with other consecutive terms

\[

$$\begin{aligned} \frac{f(1)}{f(0)} &= \frac{2}{6} = \frac{1}{3} \\ \frac{f(2)}{f(1)} &= \frac{2/3}{2} = \frac{1}{3} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-correct>(A) \(\frac{1}{3}\)</mcq-correct>
<mcq-option>(B) \(\frac{2}{3}\)</mcq-option>
<mcq-option>(C) 2</mcq-option>
<mcq-option>(D) 6</mcq-option>
</answer>

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"question_type": "Multiple Choice",
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Answer:

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</pre_analysis>

<reasoning>

Identify the exponential function form

\[
f(x) = a \cdot b^x
\]

Calculate the ratio of consecutive terms

\[
b = \frac{f(x+1)}{f(x)} = \frac{f(0)}{f(-1)} = \frac{6}{18} = \frac{1}{3}
\]

Verify with other consecutive terms

\[

$$\begin{aligned} \frac{f(1)}{f(0)} &= \frac{2}{6} = \frac{1}{3} \\ \frac{f(2)}{f(1)} &= \frac{2/3}{2} = \frac{1}{3} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-correct>(A) \(\frac{1}{3}\)</mcq-correct>
<mcq-option>(B) \(\frac{2}{3}\)</mcq-option>
<mcq-option>(C) 2</mcq-option>
<mcq-option>(D) 6</mcq-option>
</answer>

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"question_type": "Multiple Choice",
"knowledge_point": [
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