HttpTrigger.cs

using System.Threading.Tasks;
using Microsoft.AspNetCore.Mvc;
using Microsoft.Azure.WebJobs;
using Microsoft.Azure.WebJobs.Extensions.Http;
using Microsoft.AspNetCore.Http;
using Microsoft.Extensions.Logging;

namespace DIExample
{
    public class HttpTrigger
    {
        private readonly IMyService _myService;

        public HttpTrigger(IMyService myService)
        {
            _myService = myService;
        }

        [FunctionName("MyHttpTrigger")]
        public async Task<IActionResult> Run(
            [HttpTrigger(AuthorizationLevel.Anonymous, "get", Route = null)] HttpRequest req,
            ILogger log)
        {
            // Call the service that the dependency injection system gave us
            await _myService.DoSomethingAsync();

            return new NoContentResult();
        }
    }
}

nuget.sh

dotnet add package Microsoft.Extensions.DependencyInjection
dotnet add package Microsoft.Azure.Functions.Extensions

Startup.cs

using Microsoft.Azure.Functions.Extensions.DependencyInjection;
using Microsoft.Extensions.DependencyInjection;

[assembly: FunctionsStartup(typeof(DIExample.Startup))]

namespace DIExample
{
    public class Startup : FunctionsStartup
    {
        public override void Configure(IFunctionsHostBuilder builder)
        {
            builder.Services.AddTransient<IMyService, MyService>();
        }
    }
}