2002 AMC 12A Problems/Problem 19

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Problem

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

$\text{(A) }2\qquad\text{(B) }4\qquad\text{(C) }5\qquad\text{(D) }6\qquad\text{(E) }7$

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 $2+4=\boxed{6}$ solutions.

2002 AMC 12A (Problems • Resources)
Preceded by Problem 18	Followed by Problem 20
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2002 AMC 12A Problems/Problem 19

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