Question

which function represents exponential growth?

\\( f(x) = 3x \\)
\\( f(x) = x^3 \\)
\\( f(x) = x + 3 \\)
\\( f(x) = 3^x \\)

Explanation:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [
"Exponential Functions"
],
"new_concepts": [],
"current_concepts": [
"Exponential Functions",
"Exponential Growth"
]
}
</pre_analysis>

<reasoning>

Identify the definition of an exponential growth function

An exponential growth function is of the form \(f(x) = a \cdot b^x\), where \(a > 0\) and \(b > 1\).

Analyze each given option

• \(f(x) = 3x\): This is a linear function.
• \(f(x) = x^3\): This is a cubic (polynomial) function.
• \(f(x) = x + 3\): This is a linear function.
• \(f(x) = 3^x\): This is an exponential function with base \(b = 3 > 1\), representing exponential growth.

</reasoning>

<answer>
<mcq-option>(A) \(f(x) = 3x\)</mcq-option>
<mcq-option>(B) \(f(x) = x^3\)</mcq-option>
<mcq-option>(C) \(f(x) = x + 3\)</mcq-option>
<mcq-correct>(D) \(f(x) = 3^x\)</mcq-correct>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Exponential Functions"
]
}
</post_analysis>

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [
"Exponential Functions"
],
"new_concepts": [],
"current_concepts": [
"Exponential Functions",
"Exponential Growth"
]
}
</pre_analysis>

<reasoning>

Identify the definition of an exponential growth function

An exponential growth function is of the form \(f(x) = a \cdot b^x\), where \(a > 0\) and \(b > 1\).

Analyze each given option

• \(f(x) = 3x\): This is a linear function.
• \(f(x) = x^3\): This is a cubic (polynomial) function.
• \(f(x) = x + 3\): This is a linear function.
• \(f(x) = 3^x\): This is an exponential function with base \(b = 3 > 1\), representing exponential growth.

</reasoning>

<answer>
<mcq-option>(A) \(f(x) = 3x\)</mcq-option>
<mcq-option>(B) \(f(x) = x^3\)</mcq-option>
<mcq-option>(C) \(f(x) = x + 3\)</mcq-option>
<mcq-correct>(D) \(f(x) = 3^x\)</mcq-correct>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Exponential Functions"
]
}
</post_analysis>