we use AF on service plan and don't have such issue. Did you try to check whether slowness caused by Azure runtime or logic inside AF itself? E.g. if to add trace log as the first line in AF:
public static class MyFunction
{
[FunctionName("MyFunction")]
public static async Task<HttpResponseMessage> Run([HttpTrigger(AuthorizationLevel.Anonymous, "post", Route = null)]HttpRequestMessage req, TraceWriter log)
{
log.Info("Start MyFunction");
...
}
}
then open AF log (Function app > Functions > MyFunction > Code + Test > Log) and invoke your function - will this trace appear with delay or fast?
Another thing I would try is to create new Function app with regular consumption plan and try to invoke function from there. Will this problem still be there?