| 模型 | 模型答案 | 正确答案 |
|---|---|---|
| 金声玉亮2.0 | To solve the problem, we need to determine the value of \(2 \cdot g(6)\) given the functions \(f(2x) = x + 5\) and \(f(g(6)) = 13\). First, let's analyze the function \(f(2x) = x + 5\). This tells us that for any input \(2x\), the output of the function \(f\) is \(x + 5\). Next, we are given that \(f(g(6)) = 13\). We need to find \(g(6)\) such that \(f(g(6)) = 13\). From the function \(f(2x) = x + 5\), we can rewrite it in terms of \(f(y)\) where \(y = 2x\). Thus, \(f(y) = \frac{y}{2} + 5\). Now, we know \(f(g(6)) = 13\). Substituting \(g(6)\) into the function \(f\), we get: \[ f(g(6)) = \frac{g(6)}{2} + 5 = 13 \] To find \(g(6)\), we solve the equation: \[ \frac{g(6)}{2} + 5 = 13 \] Subtract 5 from both sides: \[ \frac{g(6)}{2} = 8 \] Multiply both sides by 2: \[ g(6) = 16 \] Now, we need to find \(2 \cdot g(6)\): \[ 2 \cdot g(6) = 2 \cdot 16 = 32 \] Thus, the answer is: \[ \boxed{C} \] |
C |
| deepseekr1 | To solve for \(2 \cdot g(6)\), start by expressing \(f(y)\) in terms of \(y\). Given \(f(2x) = x + 5\), let \(y = 2x\), which implies \(x = \frac{y}{2}\). Substituting, we get \(f(y) = \frac{y}{2} + 5\). Next, use the given \(f(g(6)) = 13\) and substitute \(g(6)\) into \(f\): \[ f(g(6)) = \frac{g(6)}{2} + 5 = 13. \] Solving for \(g(6)\): \[ \frac{g(6)}{2} = 13 - 5 = 8 \implies g(6) = 16. \] Finally, calculate \(2 \cdot g(6)\): \[ 2 \cdot 16 = 32. \] **Answer:** C. 32 |
C |
| gpto1 | C | C |
| GPT-4o | To determine the value of \( 2 \cdot g(6) \), we need to solve for \( g(6) \) first. We start with the given function \( f(2x) = x + 5 \). We need to find \( f(g(6)) = 13 \). Let's denote \( g(6) \) by some variable, say \( y \). Therefore, we have: \[ f(y) = 13 \] Now, according to the problem, \( f(2x) = x + 5 \). To use this, we need to rewrite \( f(y) \) in terms of \( f(2x) \). Notice that if we let \( y = 2a \), then: \[ f(2a) = a + 5 \] Given \( f(y) = 13 \) and substituting \( y = 2a \), we get: \[ f(2a) = 13 \] From the function definition \( f(2a) = a + 5 \), we equate it to 13: \[ a + 5 = 13 \] Solving for \( a \): \[ a = 13 - 5 \] \[ a = 8 \] Since \( y = 2a \), we have: \[ y = 2 \cdot 8 \] \[ y = 16 \] So \( g(6) = 16 \). Finally, we need to find \( 2 \cdot g(6) \): \[ 2 \cdot g(6) = 2 \cdot 16 \] \[ 2 \cdot g(6) = 32 \] Thus, the answer is: \[ \boxed{C} \] |
C |
| llama3.1-70B-Instruct | To solve this problem, we need to find the value of g(6) first. Given that f(g(6)) = 13, we can substitute g(6) into the function f(2x) = x + 5. Let's say g(6) = y. Then, f(2y) = y + 5. Since f(g(6)) = 13, we can set up the equation: y + 5 = 13. Solving for y, we get: y = 8. Now, we know that g(6) = 8. The question asks for 2 * g(6), so we multiply 2 by 8: 2 * g(6) = 2 * 8 = 16. The correct answer is B. |
C |
| llama3.1-405B-Instruct | B. 16 |
C |
| DeepSeek-Coder-V2 | To solve the problem, we need to determine the value of \(2 \cdot g(6)\) given the functions \(f(2x) = x + 5\) and \(f(g(6)) = 13\). First, let's analyze the function \(f(2x) = x + 5\). This tells us that for any input \(2x\), the output of the function \(f\) is \(x + 5\). Next, we are given that \(f(g(6)) = 13\). We need to find \(g(6)\) such that when \(g(6)\) is substituted into \(f\), the result is 13. From the function \(f(2x) = x + 5\), we can rewrite it in terms of \(f(y)\) where \(y = 2x\). Thus, \(f(y) = \frac{y}{2} + 5\). Now, we know \(f(g(6)) = 13\). Substituting \(g(6)\) into the function \(f\), we get: \[ f(g(6)) = \frac{g(6)}{2} + 5 = 13 \] To find \(g(6)\), we solve the equation: \[ \frac{g(6)}{2} + 5 = 13 \] Subtract 5 from both sides: \[ \frac{g(6)}{2} = 8 \] Multiply both sides by 2: \[ g(6) = 16 \] Now, we need to find \(2 \cdot g(6)\): \[ 2 \cdot g(6) = 2 \cdot 16 = 32 \] Thus, the correct answer is: \[ \boxed{C} \] |
C |
| Qwen2-72B-Instruct | To solve this problem, we need to find the value of \(g(6)\) first. We know that \(f(g(6)) = 13\). Given that \(f(2x) = x + 5\), we can set \(2x = g(6)\) and \(f(g(6)) = 13\). So, \(f(2x) = x + 5 = 13\). Solving for \(x\), we get \(x = 8\). Since \(2x = g(6)\), then \(g(6) = 2 * 8 = 16\). The question asks for \(2 * g(6)\), which is \(2 * 16 = 32\). Answer: C. 32 |
C |
| baichuan4 | First, let's find g(6) by using the given information f(2x) = x + 5 and f(g(6)) = 13. f(g(6)) = 13 g(6) + 5 = 13 (since f(2x) = x + 5, replace 2x with g(6)) g(6) = 8 Now, we need to find 2 * g(6): 2 * g(6) = 2 * 8 = 16 Answer: B. 16 |
C |
| doubao | Let y = 2x, then x = y/2. Substituting into f(2x) = x + 5 gives f(y) = y/2 + 5. Since f(g(6)) = 13, let g(6) = y, then f(y) = y/2 + 5 = 13, y/2 = 13 - 5 = 8, y = 16, so g(6) = 16, and 2 * g(6) = 2 * 16 = 32. So the answer is C. |
C |
| minimax | Let's solve the problem step by step. First, we are given that f(2x) = x + 5. We need to find the value of g(6) such that f(g(6)) = 13. Since f(g(6)) = 13, we can set up the equation using the given function form: f(2 * g(6)) = g(6) + 5 = 13 Now, we solve for g(6): g(6) + 5 = 13 g(6) = 13 - 5 g(6) = 8 Now we need to find 2 * g(6): 2 * g(6) = 2 * 8 = 16 Therefore, the answer is: B. 16 |
C |
| kimi | Let's solve the problem step by step. Given that \( f(2x) = x + 5 \), we can find the function \( f(x) \) by substituting \( x \) with \( \frac{y}{2} \), which gives us \( f(y) = \frac{y}{2} + 5 \). Now, we are given that \( f(g(6)) = 13 \). Using the function we found, we can write this as: \[ f(g(6)) = \frac{g(6)}{2} + 5 = 13 \] Now, let's solve for \( g(6) \): \[ \frac{g(6)}{2} = 13 - 5 \] \[ \frac{g(6)}{2} = 8 \] \[ g(6) = 8 \times 2 \] \[ g(6) = 16 \] Now we need to find \( 2 \times g(6) \): \[ 2 \times g(6) = 2 \times 16 \] \[ 2 \times g(6) = 32 \] So the answer is: C. 32 |
C |
| wenxinyiyan | B Explanation: Given $f(2x) = x + 5$, Let $2x = u$, then $x = \frac{u}{2}$. So, $f(u) = \frac{u}{2} + 5$. Now, given $f(g(6)) = 13$, Substitute $f(u)$ into this equation, we get: $\frac{g(6)}{2} + 5 = 13$. Solve this equation for $g(6)$, we have: $\frac{g(6)}{2} = 8$, $g(6) = 16$. Finally, $2 \times g(6) = 2 \times 16 = 32$. However, the question asks for $2 * g(6)$ directly, which is $2 * 16 = 32$. But since the options are given as the value of $g(6)$, the answer should match one of the options after dividing by 2. Therefore, the answer is B, which is $16$ (since $32 / 2 = 16$). Note: There seems to be a slight ambiguity in the question. If the question meant to ask the final value of $2 * g(6)$ after calculating $g(6)$, then the answer would be $32$. But based on the options provided, it seems the question is asking for the value of $g(6)$ itself, which is $16$. |
C |
| zhipu | To solve this problem, we need to use the given information about the function \( f \) and the relationship between \( f \) and \( g \). First, we have the function \( f(2x) = x + 5 \). We also know that \( f(g(6)) = 13 \). Let's set \( y = g(6) \). Then, we have \( f(y) = 13 \). Now, we need to express \( f(y) \) in terms of the given function \( f(2x) = x + 5 \). Since \( y \) is a general variable, we need to match it to the form \( 2x \). Let's rewrite \( f(y) \) using the given function: - If \( y = 2x \), then \( f(y) = f(2x) = x + 5 \). To find \( y \), we need to match \( f(y) = 13 \) with \( x + 5 \): \[ x + 5 = 13 \] \[ x = 8 \] Since \( y = 2x \), we substitute \( x = 8 \): \[ y = 2 \cdot 8 = 16 \] Therefore, \( g(6) = 16 \). Now, we need to find \( 2 \cdot g(6) \): \[ 2 \cdot g(6) = 2 \cdot 16 = 32 \] So, the correct answer is: C. 32 |
C |
| gpto1 | C | C |
| deepseekv3 | To solve the problem, follow these steps: 1. **Find the function \( f(x) \):** Given \( f(2x) = x + 5 \), let \( u = 2x \). Then \( x = \frac{u}{2} \), so: \[ f(u) = \frac{u}{2} + 5. \] Thus, \( f(x) = \frac{x}{2} + 5 \). 2. **Find \( g(6) \):** Given \( f(g(6)) = 13 \), substitute into \( f(x) \): \[ \frac{g(6)}{2} + 5 = 13. \] Solve for \( g(6) \): \[ \frac{g(6)}{2} = 8 \quad \Rightarrow \quad g(6) = 16. \] 3. **Compute \( 2 \cdot g(6) \):** \[ 2 \cdot g(6) = 2 \cdot 16 = 32. \] **Answer:** \(\boxed{C}\) |
C |