Problem 12 · 2015 Math Kangaroo
Hard
Geometry & Measurement
careful-counting
The x-axis and the graphs of \(f(x) = 2 - x^2\) and \(g(x) = x^2 - 1\) split the coordinate plane into
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Answer: D — 10 regions
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Hint 1 of 2
First locate where the two parabolas meet each other and where each meets the x-axis.
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Hint 2 of 2
Each new curve added to the plane can cut existing regions into more pieces.
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Approach: count regions made by the three curves
- The three curves meet in six distinct points: the down-parabola y = 2−x² hits the x-axis at \(x=\pm\sqrt2\), the up-parabola y = x²−1 hits it at \(x=\pm1\), and the two parabolas meet at \(x=\pm\sqrt{3/2}\).
- Sketching all three through those crossings and counting every separate piece of the plane (the bounded slivers between the curves plus the unbounded outer regions) gives 10 regions (D).
Mark:
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