Proving a Function is Bijective

Intro to Proofs · Axiom Academy

EXAMPLE Proving a Function is Bijective Work through a complete two-part proof that a specific function is both injective and surjective. Prove that the function defined by f(x) = 2x + 3 is bijective . To do this we will show it is both injective (one-to-one) and surjective (onto). Nice work — you proved that f(x) = 2x + 3 is bijective. Here is what the two-part proof relied on: Bijective = injective + surjective: a function must be both one-to-one and onto to be a bijection. Proving injectivity: assume f(a) = f(b) and use algebra to force a = b , showing no two different inputs share an output. Proving surjectivity: for an arbitrary y in the codomain, solve f(x) = y to construct a pre-image x in the domain. Linear functions: any f(x) = mx + b with from to is bijective by exactly this argument. Always verify: after finding the pre-image formula, substitute it back to confirm . This injective-plus-surjective structure is the standard template for proving any function is a bijection — you will reuse it throughout your studies.

This is the written version of the interactive lesson above. See the full Intro to Proofs course.