Inheritance and innovation play important roles in the conceptual design phase. Inheritance can achieve rapid design by reusing existing design knowledge, while innovation can realize innovative design by developing creative ideas and concepts. Thus, an ideal conceptual design model should administer to the implementation of inheritance and innovation. Considering rational design domains and mapping relations is beneficial to knowledge representation and inheritance. According with the thinking process of humans it is propitious to human intervention and innovation. To meet these requirements, this article proposes a hierarchical functional solving framework with hybrid mappings. In this framework, there are four design domains including function, working principle, behaviour and structure. The reasoning process of product function design consists of four mapping patterns, which are solving mapping pattern, reformulation mapping pattern, decomposition mapping pattern and derivative mapping pattern, totalling 15 basic mapping relations. The evolutionary logic of these mapping relations is developed in this article. In addition, the flow of the solving process from each design domain is established. The procedure in applying the proposed framework to the conceptual design is presented subsequently. Moreover, several embodiment strategies ensuring the effectiveness of identified design solutions are also proposed, including performance match analysis, behavioural compatibility analysis and structural compatibility analysis. Finally, a design case of the telescopic shuttle mechanism of an automatic storage/retrieval machine is studied to illustrate and demonstrate the feasibility of the proposed framework.