5G mobile networks are envisioned to sub- stantiate new vertical services with diverse performance requirements. Slicing in the Radio Access Network (RAN) promises an efficient solution for these diversified needs of 5G networks, which foresees the separation of the Base Station (BS) functionality between the Central Unit (CU) and the distributed Remote Radio Heads (RRHs). In this paper, we formulate a Mixed Integer Programming (MIP) framework that maximizes the throughput by jointly selecting the optimal Functional Split (FS) and the routing path from a connected User Equipment (UE) to the CU, while satisfying the agreed Service Level Agreements (SLAs) of each service. Furthermore, we propose an effective heuristic, SlicedRAN, which creates isolated RAN slices premised on the ser- vice requirements connected through a Fronthaul/Backhaul (FH/BH) network and obtains near-optimal solutions in a short computing time compared to the MIP framework. Our results show that there is a trade-off between the architecture of the FH/BH network and the minimum SLA of each slice, which provides a solution to efficiently design a virtualized network infrastructure. According to the results, the SlicedRAN outperforms existing State-of-the-Art (SoA) up to 112% gain in throughput. Results are shown close to the optimal results, with a loss below 5%.