| 模型 | 模型答案 | 正确答案 |
|---|---|---|
| 金声玉亮2.0 | The function $h(x)$ is a composition of $f$ and $g$. To find the inverse of $h(x)$, we need to find the inverse of the composition, which is the composition of the inverses in reverse order. First, let's find the inverse of $g(x) = 2x - 3$. To do this, we can swap $x$ and $g(x)$ and solve for $g(x)$: $x = 2g(x) - 3$ $g(x) = \frac{x + 3}{2}$ So, the inverse of $g(x)$ is $g^{-1}(x) = \frac{x + 3}{2}$. Next, let's find the inverse of $f(x) = 3x + 4$. Again, we can swap $x$ and $f(x)$ and solve for $f(x)$: $x = 3f(x) + 4$ $f(x) = \frac{x - 4}{3}$ So, the inverse of $f(x)$ is $f^{-1}(x) = \frac{x - 4}{3}$. Now, we can find the inverse of $h(x)$ by composing the inverses of $f$ and $g$ in reverse order: $h^{-1}(x) = g^{-1}(f^{-1}(x))$ $h^{-1}(x) = g^{-1}\left(\frac{x - 4}{3}\right)$ $h^{-1}(x) = \frac{\frac{x - 4}{3} + 3}{2}$ $h^{-1}(x) = \frac{x - 4 + 9}{6}$ $h^{-1}(x) = \frac{x + 5}{6}$ Therefore, the inverse of $h(x)$ is $\frac{x + 5}{6}$. The correct answer is C. |
C |
| deepseekr1 | To find the inverse of \( h(x) = f(g(x)) \), first compute the composition. Given \( f(x) = 3x + 4 \) and \( g(x) = 2x - 3 \): \[ h(x) = f(g(x)) = 3(2x - 3) + 4 = 6x - 9 + 4 = 6x - 5. \] Next, find the inverse of \( h(x) = 6x - 5 \). Start by swapping \( x \) and \( y \): \[ x = 6y - 5. \] Solve for \( y \): \[ x + 5 = 6y \implies y = \frac{x + 5}{6}. \] Thus, the inverse function is \( \frac{x + 5}{6} \), which corresponds to option **C**. **Answer:** C. \(\frac{x+5}{6}\) |
C |
| gpto1 | C | C |
| GPT-4o | To find the inverse of \( h(x) \), we first need to determine \( h(x) \). Given \( f(x) = 3x + 4 \) and \( g(x) = 2x - 3 \), we express \( h(x) \) as follows: \[ h(x) = f(g(x)) \] Substitute \( g(x) \) into \( f(x) \): \[ h(x) = f(2x - 3) = 3(2x - 3) + 4 \] \[ h(x) = 3 \cdot 2x - 3 \cdot 3 + 4 \] \[ h(x) = 6x - 9 + 4 \] \[ h(x) = 6x - 5 \] Now, we need to find the inverse of \( h(x) = 6x - 5 \). Let \( y = h(x) \): \[ y = 6x - 5 \] To find the inverse, solve for \( x \) in terms of \( y \): \[ y = 6x - 5 \] \[ y + 5 = 6x \] \[ x = \frac{y + 5}{6} \] Thus, the inverse function \( h^{-1}(x) \) is: \[ h^{-1}(x) = \frac{x + 5}{6} \] Therefore, the correct answer is: C. \(\frac{x+5}{6}\) |
C |
| llama3.1-70B-Instruct | ## Step 1: First, we need to find the composite function $h(x)=f(g(x))$. This means we substitute $g(x)$ into $f(x)$. $h(x) = f(g(x)) = 3(2x-3) + 4$ ## Step 2: Simplify the expression for $h(x)$. $h(x) = 6x - 9 + 4 = 6x - 5$ ## Step 3: To find the inverse of $h(x)$, we need to swap $x$ and $y$ and then solve for $y$. Let $y = 6x - 5$. Swapping gives $x = 6y - 5$. ## Step 4: Solve for $y$. $x = 6y - 5 \Rightarrow 6y = x + 5 \Rightarrow y = \frac{x + 5}{6}$ ## Step 5: The inverse of $h(x)$ is $\frac{x + 5}{6}$. The final answer is: $\boxed{C}$ |
C |
| llama3.1-405B-Instruct | The correct answer is C |
C |
| DeepSeek-Coder-V2 | To find the inverse of \( h(x) \), we first need to determine \( h(x) \) by composing \( f(x) \) and \( g(x) \). Given: \[ f(x) = 3x + 4 \] \[ g(x) = 2x - 3 \] We need to find \( h(x) = f(g(x)) \). First, compute \( g(x) \): \[ g(x) = 2x - 3 \] Next, substitute \( g(x) \) into \( f(x) \): \[ h(x) = f(g(x)) = f(2x - 3) \] Now, substitute \( 2x - 3 \) into \( f(x) \): \[ f(2x - 3) = 3(2x - 3) + 4 \] \[ = 6x - 9 + 4 \] \[ = 6x - 5 \] So, \( h(x) = 6x - 5 \). To find the inverse of \( h(x) \), we need to solve for \( x \) in terms of \( y \) where \( y = h(x) \): \[ y = 6x - 5 \] Rearrange to solve for \( x \): \[ y + 5 = 6x \] \[ x = \frac{y + 5}{6} \] Thus, the inverse function \( h^{-1}(x) \) is: \[ h^{-1}(x) = \frac{x + 5}{6} \] Therefore, the correct answer is: \[ \boxed{C} \] |
C |
| Qwen2-72B-Instruct | D. \frac{x-5}{6} | C |
| baichuan4 | To find the inverse of $h(x)$, we first need to determine $h(x)$ by composing $f(x)$ with $g(x)$. Given: $f(x) = 3x + 4$ $g(x) = 2x - 3$ Composing $f$ with $g$ gives us $h(x)$: $h(x) = f(g(x)) = 3(2x - 3) + 4 = 6x - 9 + 4 = 6x - 5$ Now, to find the inverse of $h(x)$, we need to solve the equation $h(x) = y$ for $x$ in terms of $y$. $6x - 5 = y$ $6x = y + 5$ $x = \frac{y + 5}{6}$ Replacing $y$ with $x$ (since when finding the inverse, we switch the roles of $x$ and $y$), we get: $x = \frac{x + 5}{6}$ Therefore, the inverse of $h(x)$ is $\frac{x + 5}{6}$. Answer: C |
C |
| doubao | First find $h(x)$: $h(x)=f(g(x))=f(2x-3)=3(2x-3)+4=6x-9+4=6x-5$. To find the inverse of $h(x)$, let $y=6x-5$, then $6x=y+5$, $x=\frac{y+5}{6}$. So the inverse of $h(x)$ is $\frac{x+5}{6}$. So the answer is C. |
C |
| minimax | To find the inverse of $h(x)$, we first need to find the expression for $h(x)$ by composing $f(x)$ and $g(x)$: $h(x) = f(g(x)) = f(2x - 3) = 3(2x - 3) + 4 = 6x - 9 + 4 = 6x - 5$ Now, to find the inverse function $h^{-1}(x)$, we need to solve the equation $h(x) = y$ for $x$: $y = 6x - 5$ Add 5 to both sides: $y + 5 = 6x$ Divide both sides by 6: $\frac{y + 5}{6} = x$ Now, to express $h^{-1}(x)$, we replace $y$ with $x$: $h^{-1}(x) = \frac{x + 5}{6}$ So the correct answer is: C. $\frac{x+5}{6}$ |
C |
| kimi | First, let's find the function \( h(x) \) by substituting \( g(x) \) into \( f(x) \): \[ h(x) = f(g(x)) = f(2x - 3) = 3(2x - 3) + 4 \] Now, let's simplify \( h(x) \): \[ h(x) = 6x - 9 + 4 = 6x - 5 \] To find the inverse of \( h(x) \), we set \( y = h(x) \) and solve for \( x \): \[ y = 6x - 5 \] Now, solve for \( x \): \[ y + 5 = 6x \] \[ x = \frac{y + 5}{6} \] So, the inverse function \( h^{-1}(x) \) is: \[ h^{-1}(x) = \frac{x + 5}{6} \] The correct answer is C. \(\boxed{\frac{x+5}{6}}\) |
C |
| wenxinyiyan | D | C |
| zhipu | To find the inverse of \( h(x) \), we first need to determine the expression for \( h(x) \). Given: \[ f(x) = 3x + 4 \] \[ g(x) = 2x - 3 \] We know: \[ h(x) = f(g(x)) \] First, we compute \( g(x) \): \[ g(x) = 2x - 3 \] Next, we substitute \( g(x) \) into \( f(x) \): \[ h(x) = f(g(x)) = f(2x - 3) \] Now, substitute \( 2x - 3 \) into \( f(x) \): \[ f(2x - 3) = 3(2x - 3) + 4 \] \[ = 6x - 9 + 4 \] \[ = 6x - 5 \] So, \( h(x) = 6x - 5 \). To find the inverse of \( h(x) \), we need to solve for \( x \) in the equation \( y = 6x - 5 \). Rewrite the equation: \[ y = 6x - 5 \] Add 5 to both sides: \[ y + 5 = 6x \] Divide by 6: \[ x = \frac{y + 5}{6} \] Thus, the inverse function \( h^{-1}(x) \) is: \[ h^{-1}(x) = \frac{x + 5}{6} \] Therefore, the correct answer is: C. \(\frac{x + 5}{6}\) |
C |
| gpto1 | C | C |
| deepseekv3 | To find the inverse of \( h(x) \), we first determine \( h(x) \) by composing \( f \) and \( g \): \[ h(x) = f(g(x)) \] Given: \[ f(x) = 3x + 4 \quad \text{and} \quad g(x) = 2x - 3 \] Substitute \( g(x) \) into \( f(x) \): \[ h(x) = f(g(x)) = 3(2x - 3) + 4 = 6x - 9 + 4 = 6x - 5 \] Now, to find the inverse \( h^{-1}(x) \), solve for \( x \) in the equation \( y = 6x - 5 \): \[ y = 6x - 5 \\ y + 5 = 6x \\ x = \frac{y + 5}{6} \] Thus, the inverse function is: \[ h^{-1}(x) = \frac{x + 5}{6} \] The correct answer is **C**. \boxed{C} |
C |