2002 AMC 12A Problem 19

Attempt Problem 19 of the 2002 AMC 12A below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2002 AMC 12A solutions, or check the answer key.

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19.

The graph of the function

f

is shown below. How many solutions does the equation

f(f(x)) = 6

have?

2

4

5

6

7

f(f(x)) = 6

means

f(x)

must be an input where

f

equals

6

From the graph

f(t) = 6

t = -2

and

t = 1;

count solutions of

f(x) = -2

and

f(x) = 1

The graph reaches

6

x = -2

and

x = 1,

f(f(x)) = 6

requires

f(x) = -2

f(x) = 1.

The horizontal line

y = -2

meets the graph twice, and

y = 1

meets it four times, giving

2 + 4 = 6

solutions.

Thus, the correct answer is D.

Problem 19 in other years