Problem

\( \mathrm{Q} \) 函数图像 能描述函数图像的趋向 Let \( f(x)=p(x) \wedge \operatorname{left}\left(x^{\wedge}\{\right. \) wedge \( \} \) 2-7 \( x+12 \) | right), where \( p(x) \) is a polynomial. Which is a possible graph of \( y=f(x) \) ? To describe the graph of the function \( f(x) \), we need to analyze its behavior and find its key features. Here are the steps to solve this problem:

Solution

Step 1 :首先观察函数的定义,可以看出 f(x) 是由两个函数取“最小值”而得,即 f(x) = min{p(x), x^2-7x+12}。

Step 2 :根据二次函数的性质,易得 x^2-7x+12 = (x-4)(x-3) 的零点为 4 和 3。

Step 3 :因此,x ∈ (-∞,3) 时,f(x) = p(x);x ∈ [3,4] 时,f(x) = x^2-7x+12;x ∈ (4,+∞) 时,f(x) = p(x)。

Step 4 :综上所述,函数 f(x) 具有下列几个特征:

Step 5 :当 x → -∞ 时,f(x) → -∞。

Step 6 :当 x → +∞ 时,f(x) → +∞。

Step 7 :当 x ∈ (-∞,3) 时,f(x) = p(x)。

Step 8 :当 x ∈ [3,4] 时,f(x) = x^2-7x+12。

Step 9 :当 x ∈ (4,+∞) 时,f(x) = p(x)。

Step 10 :根据函数的特点,可以得出选项中有可能是函数 f(x) 的图像的是 D。

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Source: https://solvelyapp.com/problems/8819/

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