Finding Inverse Functions

Discrete Math · Axiom Academy

EXAMPLE Finding Inverse Functions Learn to find inverse functions algebraically and verify composition properties Not all functions have inverses! A function must be bijective (both one-to-one and onto) to be invertible. Click on the functions that are NOT invertible : Fails horizontal line test (not one-to-one) Passes horizontal line test (bijective) Multiple inputs map to same output (not one-to-one) Each output has exactly one input (bijective) Excellent work! You've learned how to find and verify inverse functions. Here's what we covered: Finding Inverses: Replace f(x) with y, swap x and y, then solve for y to get f⁻¹(x) Verification: A function and its inverse satisfy (f ∘ f⁻¹)(x) = x and (f⁻¹ ∘ f)(x) = x Bijective Requirement: Only bijective functions (one-to-one and onto) have inverses Horizontal Line Test: A function has an inverse if and only if every horizontal line intersects the graph at most once Non-Invertible Functions: Functions like f(x) = x² (on all reals) fail because multiple inputs map to the same output Understanding inverse functions is crucial in discrete mathematics for studying bijections, permutations, and cryptographic functions where reversibility is essential!

This is the written version of the interactive lesson above. See the full Discrete Math course.