AMC12 2002 A

AMC12 2002 A · Q19

AMC12 2002 A · Q19. It mainly tests Functions basics, Algebra misc.

The graph of the function $f$ is shown below. How many solutions does the equation $f(f(x))=6$ have?

函数 $f$ 的图像如下所示。方程 $f(f(x))=6$ 有多少个解？

(A) 2 2

(B) 4 4

(D) 6 6

(E) 7 7

Answer

Correct choice: (D)

正确答案：（D）

Solution

First of all, note that the equation $f(t)=6$ has two solutions: $t=-2$ and $t=1$. Given an $x$, let $f(x)=t$. Obviously, to have $f(f(x))=6$, we need to have $f(t)=6$, and we already know when that happens. In other words, the solutions to $f(f(x))=6$ are precisely the solutions to ($f(x)=-2$ or $f(x)=1$). Without actually computing the exact values, it is obvious from the graph that the equation $f(x)=-2$ has two and $f(x)=1$ has four different solutions, giving us a total of $2+4=\boxed{(D)6}$ solutions.

首先注意到方程 $f(t)=6$ 有两个解：$t=-2$ 和 $t=1$。给定一个 $x$，令 $f(x)=t$。要使 $f(f(x))=6$，必须有 $f(t)=6$，而我们已知这在何时发生。换句话说，$f(f(x))=6$ 的解恰好是满足（$f(x)=-2$ 或 $f(x)=1$）的所有 $x$。不必计算精确值，从图像可直接看出方程 $f(x)=-2$ 有两个解，而 $f(x)=1$ 有四个不同的解，因此总共有 $2+4=\boxed{(D)6}$ 个解。

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