Problem 18 · 2014 Math Kangaroo
Hard
Algebra & Patterns
custom-operationsubstitution
The function \(f(x)=ax+b\) fulfils the conditions \(f(f(f(1)))=29\) and \(f(f(f(0)))=2\). What is the value of \(a\)?
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Answer: C — 3
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Hint 1 of 2
Compose f three times: f(f(f(x))) is again a linear function a³x + (something).
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Hint 2 of 2
Subtract the two given equations so the constant part cancels.
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Approach: compose, then subtract to isolate a³
- For f(x)=ax+b, applying it three times gives f(f(f(x))) = a³x + b(a²+a+1).
- Then f(f(f(1))) − f(f(f(0))) = a³ = 29 − 2 = 27.
- So a³ = 27 and a = 3.
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