Problem 21 · 2012 Math Kangaroo
Stretch
Algebra & Patterns
substitution
Which of the following functions fulfills for all x ≠ 0 the condition \(f\!\left(\tfrac{1}{x}\right) = f(x)\)?
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Answer: E — \(f(x) = x + \tfrac{1}{x}\)
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Hint 1 of 2
Replace \(x\) by \(\tfrac1x\) in each candidate and see which one comes back unchanged.
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Hint 2 of 2
A combination that is symmetric in \(x\) and \(\tfrac1x\) swaps to itself.
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Approach: test the symmetry f(1/x) = f(x)
- For \(f(x) = x + \tfrac1x\), replacing \(x\) by \(\tfrac1x\) gives \(\tfrac1x + x\), the same expression.
- Every other option changes value under \(x \to \tfrac1x\).
- So \(f(x) = x + \tfrac1x\) satisfies the condition, choice E.
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