Problem 22 · 2013 Math Kangaroo
Stretch
Algebra & Patterns
substitution
The function \(f : \mathbb{R} \to \mathbb{R}\) is periodic with period 5, and for \(-3 \le x < 2\) it satisfies \(f(x) = x^{2}\). What is \(f(2013)\)?
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Answer: D — 4
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Hint 1 of 2
Use the period to reduce 2013 to a number inside the defined range.
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Hint 2 of 2
Subtract multiples of 5 until you land in [−3, 2).
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Approach: reduce by the period
- 2013 = 5·402 + 3, so f(2013) = f(3) = f(3−5) = f(−2).
- −2 lies in [−3,2), where f(x)=x².
- f(−2) = 4, so D.
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