This paper presents SLATE, a fully-managed, heterogeneous Function-as-a-Service (FaaS) system for deploying serverless functions onto heterogeneous cloud infrastructures. We extend the traditional homogeneous FaaS execution model to support heterogeneous functions, automating and abstracting runtime management of heterogeneous compute resources in order to improve cloud tenant accessibility to specialised, accelerator resources, such as FPGAs and GPUs. In particular, we focus on the mechanisms required for heterogeneous scaling of deployed function instances to guarantee latency objectives while minimising cost. We develop a simulator to validate and evaluate our approach, considering case-study functions in three application domains: machine learning, bio-informatics, and physics. We incorporate empirically derived performance models for each function implementation targeting a hardware platform with combined computational capacity of 24 FPGAs and 12 CPU cores. Compared to homogeneous CPU and homogeneous FPGA functions, simulation results achieve respectively a cost improvement for non-uniform task traffic of up to 8.7 times and 1.7 times, while maintaining specified latency objectives.