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📰 Question: A philosopher of science considers a function $ k(x) $ modeling the "distance" from theory to observation, satisfying $ k(x + y) = k(x) + k(y) - 2k(xy) $. If $ k(1) = 1 $, find $ k(2) $. 📰 Solution: Set $ x = y = 1 $: $ k(2) = k(1) + k(1) - 2k(1 \cdot 1) = 1 + 1 - 2(1) = 0 $. Verify consistency: Try $ x = 2, y = 1 $: $ k(3) = k(2) + k(1) - 2k(2) = 0 + 1 - 0 = 1 $. Try $ x = y = 2 $: $ k(4) = k(2) + k(2) - 2k(4) $ → $ k(4) + 2k(4) = 0 + 0 $ → $ 3k(4) = 0 $ → $ k(4) = 0 $. Assume $ k(x) = 0 $ for all $ x $, but $ k(1) = 1 📰 eq 0 $. Contradiction? Wait, from $ k(2) = 0 $, check $ x = 1, y = -1 $: $ k(0) = k(1) + k(-1) - 2k(-1) = 1 + k(-1) - 2k(-1) = 1 - k(-1) $. Also $ k(0) = k(0 + 0) = 2k(0) - 2k(0) = 0 $? No: $ k(0) = k(0 + 0) = 2k(0) - 2k(0) = 0 $. So $ k(0) = 0 $. Then $ 0 = 1 - k(-1) $ → $ k(-1) = 1 $. Then $ x = -1, y = -1 $: $ k(-2) = 2k(-1) - 2k(1) = 2(1) - 2(1) = 0 $. $ x = 1, y = -1 $: $ k(0) = k(1) + k(-1) - 2k(-1) = 1 + 1 - 2(1) = 0 $, consistent. Now $ x = 2, y = -1 $: $ k(1) = k(2) + k(-1) - 2k(-2) = 0 + 1 - 0 = 1 $, matches. No contradiction. Thus $ k(2) = 0 $. Final answer: $ oxed{0} $. 📰 You Wont Believe Whats Locked Inside The U Of M Patient Portal 1795948 📰 Update Office Mac 615786 📰 Rudy Ruettiger 5529688 📰 Word For Weakened 4152845 📰 Integral Calculator 6577382 📰 American President During Second World War 7992415 📰 Discover The Shocking True Meaning Of Ffyou Wont Believe What It Stands For 3800737 📰 How Old Taylor Swift 3895408 📰 Wells Fargo Bank Albany Ny 1903665 📰 Cotton Candy And Candy Floss The Phenomenal Reveal You Must Witness Now 6358593 📰 Physican Npi 4997536 📰 Jon Pertwee 192237 📰 Robert Francis Prevost Education 9253410 📰 Microsoft Explore 2026 5560064 📰 Stronghold Shocked The Web The Untouched Secrets Professionals Are Ignoring 1471974