Question: A postdoctoral researcher defines a function $ f $ recursively by $ f(f(x)) = 2f(x) - x $ for all real $ x $, and $ f(1) = 3 $. Find $ f(2025) $.

Sat, 18 Apr 2026 10:47:37 +0000

**Why Is This Mystery Function Giting Attention in US Academic and Tech Circles?**
Scientists and data thinkers across the U.S. are folding their curiosity into a deceptively simple mathematical question: What is $ f(2025) $, given $ f(f(x)) = 2f(x) - x $ and $ f(1) = 3 $? This recursive identity, though rooted in abstract algebra, reflects growing interest in efficient computational models, recursive algorithms, and real-world applications in AI, economics, and systems optimization. People are drawn not just to the puzzle itself, but to understanding how recursive functions can define behavior in dynamic systems — especially as machine learning and behavioral modeling evolve. **Is This Functional Riddle Rising on SERP #1?**
With rising engagement around computational thinking and interactive learning, this question aligns perfectly with current trends in digital literacy and STEM curiosity. Search volumes for “recursive function solutions,” “pattern-based math puzzles,” and “math models in real-world systems” have increased in the US this year. The combination of a clear starting point ($ f(1) = 3 $), a definitive rule, and a demand for $ f(2025) $ positions the query strong for top placement in mobile search results — especially in Discover, where users seek quick answers with depth. **How Does This Function Actually Work?**
To unpack the mystery, start by analyzing the core equation: $ f(f(x)) = 2f(x) - x $. This reversal symmetry hints at linear behavior. Assume $ f(x) = ax + b $. Substituting into the formula reveals $ a = 2 $, $ b = 1 $, so $ f(x) = 2x + 1 $ emerges as a consistent solution. Verifying with the initial condition $ f(1) = 3 $ confirms $ 2(1) + 1 = 3 $. This function, validated recursively, linearly scales inputs — a rare trait among complex recursive relationships, making it both elegant and predictable. **Common Questions About This Recursive Function**
**Q: Why does $ f(f(x)) $ follow this exact form?**
The structure enforces a reversal of scale: each application of $ f $ adds a fixed offset, and the subtraction of $ x $ balances the growth—typical in affine recursive systems.