I cannot understand the difference between WCF (service oriented) , and Azure Function or AWS lambda ( FaaS). It seems to me both are invoking remote functions, while WCF has a host. but what is the technical difference between them?
WCF or the Windows Communication Foundation, is another framework, this time for writing and consuming services. These are either web services, or other, e.g. TCP based services, even MSMQ based services. This is, in my opinion, what you should be looking at for exposing your back-end. WCF provides you the ability to easily specify a contract and implementation, while leaving the hosting of the service and the instantiation to IIS (IIS being Microsoft's web server, which is also running under the covers on Azure).
Azure, towards you, is a hosting provider. It helps you scale your application servers based on demand (e.g. number of mobile clients downloading & installing your application).
A little marketing speak: Azure lowers your cost of ownership for your own solutions because it takes away the initial investment in firstly figuring out (guessing) the amount of hardware you need and then building/renting a data center and/or hardware. It also provides some form of middleware for your applications, like AppFabric, so that they can communicate in the "cloud" a bit better. You also get load balancing on Azure, distributed hosting (e.g. Europe datacenters, USA datacenters...), fail safe mechanism already in place (automatic instance instantiation if one were to fail) and obviously, pay as you go & what you use benefits.
Here is the reference: Introduction to Azure Functions, Azure and WCF