Problem 21 · 2021 Math Kangaroo
Stretch
Algebra & Patterns
casework
The figure shows the graph of a function \(f : [-5,5] \to \mathbb{R}\). How many distinct solutions does the equation \(f\bigl(f(x)\bigr) = 0\) have?
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Answer: E — 8
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Hint 1 of 2
First find every x where f(x)=0 from the graph.
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Hint 2 of 2
Then f(f(x))=0 means f(x) must equal one of those roots; count the x-values for each.
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Approach: solve in two layers using the roots of f
- Read the roots of f off the graph, the x-values where the curve meets the axis.
- For each root r, count how many x give f(x)=r by intersecting the graph with the horizontal line y=r.
- Adding those preimage counts over all roots gives 8 distinct solutions.
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