Autonomous race driving poses a complex control challenge as vehicles must be operated at the edge of their handling limits to reduce lap times while respecting physical and safety constraints. This paper presents a novel reinforcement learning (RL)-based approach, incorporating the action mapping (AM) mechanism to manage state-dependent input constraints arising from limited tire-road friction. A numerical approximation method is proposed to implement AM, addressing the complex dynamics associated with the friction constraints. The AM mechanism also allows the learned driving policy to be generalized to different friction conditions. Experimental results in our developed race simulator demonstrate that the proposed AM-RL approach achieves superior lap times and better success rates compared to the conventional RL-based approaches. The generalization capability of driving policy with AM is also validated in the experiments.