| 模型 | 模型答案 | 正确答案 |
|---|---|---|
| 金声玉亮2.0 | D. Domain is the same; range is tripled. Explanation: When we change sin x to 3 sin 2x, we are applying two transformations to the original function: 1. Horizontal compression by a factor of 2: This is due to the 2x inside the sine function. This transformation doubles the frequency of the function, but it does not change the domain. The domain of sin x is all real numbers, and it remains the same for 3 sin 2x. 2. Vertical stretch by a factor of 3: This is due to the coefficient 3 outside the sine function. This transformation triples the amplitude of the function, which means the range of the function is tripled. The range of sin x is [-1, 1], so the range of 3 sin 2x is [-3, 3]. Therefore, the correct answer is D: Domain is the same; range is tripled. |
D |
| deepseekr1 | The original function \( \sin x \) has a domain of all real numbers (unchanged by horizontal transformations) and a range of \([-1, 1]\). Transforming it to \( 3 \sin 2x \) involves two changes: 1. **Horizontal Compression**: The \( 2x \) argument halves the period (affects the function's repetition, not the domain). The domain remains **all real numbers**. 2. **Vertical Scaling**: The factor of 3 triples the amplitude, expanding the range to \([-3, 3]\). The range is **tripled** compared to the original. **Answer**: D. Domain is the same; range is tripled |
D |
| gpto1 | D | D |
| GPT-4o | A. Domain and range remain the same | D |
| llama3.1-70B-Instruct | D |
D |
| llama3.1-405B-Instruct | D |
D |
| DeepSeek-Coder-V2 | B. Domain is double; range remains the same | D |
| Qwen2-72B-Instruct | D. Domain is the same; range is tripled | D |
| baichuan4 | B. Domain is double; range remains the same | D |
| doubao | D The domain of both sin x and 3 sin 2x is all real numbers, so the domain remains the same. The amplitude of 3 sin 2x is 3 times that of sin x, so the range is tripled. |
D |
| minimax | A. Domain and range remain the same Explanation: The domain of the sine function is all real numbers, and this does not change when we multiply the function by a constant or when we replace x with 2x. The range of the sine function is [-1, 1], and this also remains the same because the transformation 3 sin 2x still produces values that fall within this interval. Multiplying by 3 stretches the graph vertically, but it does not affect the domain or the maximum and minimum values that the function can take. |
D |
| kimi | D. Domain is the same; range is tripled The domain of the sine function, which is \( \sin x \), is all real numbers, and changing it to \( 3 \sin 2x \) does not alter the domain because \( 2x \) can still take on all real numbers. However, the range of \( \sin x \) is \([-1, 1]\). When you multiply by 3, the range becomes \([-3, 3]\), effectively tripling the range. The function \( 2x \) inside the sine function does not change the domain but does affect the period of the function, making it twice as fast as the original sine function, but this does not affect the domain or the range in the context of this question. |
D |
| wenxinyiyan | D Explanation: 1. Domain: * The domain of $\sin x$ is all real numbers, $\mathbb{R}$. * The function $3\sin 2x$ is a transformation of $\sin x$ that involves stretching and compressing, but not shifting horizontally. Therefore, the domain remains the same: $\mathbb{R}$. 2. Range: * The range of $\sin x$ is $[-1, 1]$ because the sine function oscillates between -1 and 1. * In $3\sin 2x$, the sine function is multiplied by 3. This vertical stretch means that the maximum and minimum values of the function are now tripled. So, the new range is $[-3, 3]$. Since the domain remains the same and the range is tripled, the answer is D. |
D |
| zhipu | The transformation from \( \sin x \) to \( 3 \sin 2x \) affects both the domain and the range of the function. 1. **Domain**: The function \( \sin x \) is defined for all real numbers \( x \). The function \( \sin 2x \) is also defined for all real numbers \( x \), so the domain of \( 3 \sin 2x \) remains all real numbers. Therefore, the domain remains the same. 2. **Range**: The range of \( \sin x \) is \([-1, 1]\). For \( 3 \sin 2x \), the sine function \( \sin 2x \) still varies between \(-1\) and \(1\), but multiplying by 3 scales this range to \([-3, 3]\). Thus, the range is tripled. Based on these observations, the correct answer is: D. Domain is the same; range is tripled |
D |
| gpto1 | D | D |
| deepseekv3 | To answer this question, let's analyze the changes in the domain and range when transforming \(\sin x\) to \(3 \sin 2x\). 1. **Domain**: - The domain of \(\sin x\) is all real numbers, i.e., \((-\infty, \infty)\). - The transformation \(3 \sin 2x\) does not restrict the domain in any way, so the domain remains all real numbers, \((-\infty, \infty)\). - Thus, the **domain remains the same**. 2. **Range**: - The range of \(\sin x\) is \([-1, 1]\). - The transformation \(3 \sin 2x\) scales the amplitude of the sine function by a factor of 3. Therefore, the range becomes \([-3, 3]\). - Thus, the **range is tripled**. ### Correct Answer: **D. Domain is the same; range is tripled** |
D |