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📰 Thus, $ f(x) = rac{1}{2}x^2 + qx $, where $ q $ is arbitrary. There are infinitely many such functions. However, the original question specifies "number of functions," but the condition allows $ q \in \mathbb{R} $, leading to infinitely many solutions. If additional constraints (e.g., continuity) are implied, the solution is still infinite. But based on the structure, the answer is infinite. However, the original fragment likely intended a finite count. Revisiting, suppose the equation holds for all $ a, b $, but $ f $ is linear: $ f(x) = qx $. Substituting: $ q(a + b) = qa + qb + ab \Rightarrow 0 = ab $, which fails unless $ ab = 0 $. Thus, no linear solutions. The correct approach shows $ f(x) = rac{1}{2}x^2 + qx $, so infinitely many functions exist. But the original question may have intended a specific form. Given the context, the answer is oxed{\infty} (infinite). 📰 Kayle Aram Exposed: The Shocking Truth Behind His Rise to Fame! 📰 You Won’t Believe What Kayle Aram Revealed About His Secret Career! 📰 Glitched Out 7062458 📰 Mp Juice Transform Your Energy Overnightdrink It Twice Daily 6862387 📰 Joseph Baena Siblings 226467 📰 Saint Petersburg Florida Zip Code 8525023 📰 The Ultimate Guide Exact Ml In A Shot Glass Revealedno Guesswork 6926989 📰 Unlock Massive Returns Inside Nvidias Poolside Investment Opportunity Now 8788394 📰 The Hallway 6564833 📰 Assimil Spanish 5414382 📰 Voo 10 Year Return Shock You Wont Believe How This Investment Grew Over A Decade 3669508 📰 Pecan Tassies That Steal The Spotlight Shop Now Before Theyre Gone 7295194 📰 Ggb Stock Shock Experts Say This Single Stock Will Skyrocket Alone 5151873 📰 Stampa Nera The Dark Truth Behind This Untold Design Revolution 646122 📰 Joseph Zada Movies And Tv Shows 9090732 📰 Here I Am Lord The Shocking Lyrics To This Powerful Hymn Revealed 2881698 📰 Artifice Meaning 9510568