Problem 17 · 2013 Math Kangaroo
Medium
Algebra & Patterns
compositioncasework
Peter drew the graph of a function \(f : \mathbb{R} \to \mathbb{R}\) consisting of two rays and a line segment, as shown. How many solutions does the equation \(f(f(f(x))) = 0\) have?
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Answer: A — 4
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Hint 1 of 2
First find every input that f sends to 0, then work outward one layer at a time.
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Hint 2 of 2
Track preimages: solve f = 0, then f = those values, then again.
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Approach: count preimages layer by layer
- f(x)=0 at x = 0 and x = −4.
- Pulling back once more, f(x) ∈ {0,−4} adds x = −8; pulling back again adds x = −12.
- The solution set is {0,−4,−8,−12}: 4 solutions, A.
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