هل العكس الأيسر يعني الحقن؟

Does the concept of a left inverse in mathematics necessarily imply that a function is injective? Could you please elaborate on the relationship between these two properties and explain why or why not the presence of a left inverse guarantees injectivity? Additionally, could you provide an example or counterexample to further clarify this point? It would be helpful if you could also discuss any relevant theorems or properties that might aid in understanding this relationship.