Function Iteration Explained: Fixed Points, Induction & Limits | IB Math AA HL

What happens when you keep feeding a linear function back into itself? In this full IB Mathematics AA HL investigation, we build the formula for repeated composition, explain the geometric series, prove the result by induction, and use fixed points to understand convergence, translation and reflection. A negative multiplier can still converge when its magnitude is below one. Based on May 2025 AA HL Paper 3, TZ3, Question 1 (27 marks). fⁿ here means n-fold composition, not an ordinary power. 00:00 One rule, three futures 00:35 Composition is not a power 01:11 Compose the function twice 01:52 Why the constants form a geometric sum 02:32 The general pattern 03:14 The case m = 1 03:54 Where the geometric quotient comes from 04:38 Induction: base case and hypothesis 05:20 Induction: prove the next case 06:02 Complete the proof 06:42 Fixed points and convergence 07:26 Why an alternating sequence can still converge 08:10 The case m = −1 08:55 Conditions and final recap Try it: for f(x)=−0.6x+1.6, find the fixed point and explain why the iterates approach it. Then compare m=−1. Explore IBvia practice and expert support: https://ibvia.com/ Independent educational explanation. IBvia is not affiliated with or endorsed by the International Baccalaureate Organization.