Solution 1:

You just need to create it with schemabinding.

SQL Server will then verify whether or not it meets the criteria to be considered as deterministic (which it does as it doesn't access any external tables or use non deterministic functions such as getdate()).

You can verify that it worked with

SELECT OBJECTPROPERTY(OBJECT_ID('[dbo].[FullNameLastFirst]'), 'IsDeterministic')

Adding the schemabinding option to your original code works fine but a slightly simpler version would be.

CREATE FUNCTION [dbo].[FullNameLastFirst] (@IsPerson  BIT,
                                           @LastName  NVARCHAR(100),
                                           @FirstName NVARCHAR(100))
RETURNS NVARCHAR(201)
WITH SCHEMABINDING
AS
  BEGIN
      RETURN CASE
               WHEN @IsPerson = 0
                     OR @FirstName = '' THEN @LastName
               ELSE @LastName + ' ' + @FirstName
             END
  END

Solution 2:

You need to declare the User Defined Function WITH SCHEMABINDING to appease the 'deterministic' requirement of an index on the computed column.

A function declared WITH SCHEMABINDING will retain additional knowledge about the object dependencies used in the function (e.g. columns in the table), and will prevent any changes to these columns, unless the function itself is dropped beforehand.

Deterministic functions can also assist Sql Server in optimizing its execution plans, most notably the Halloween Protection problem.

Here's an example of creating an index on a computed column using a schema bound function:

create function [dbo].[FullNameLastFirst] 
( 
    @IsPerson bit, 
    @LastName nvarchar(100), 
    @FirstName nvarchar(100) 
) 
returns nvarchar(201) 
with schemabinding
as 
begin 
    declare @Result nvarchar(201) 
    set @Result = (case when @IsPerson = 0 then @LastName 
                        else case when @FirstName = '' then @LastName 
                                  else (@LastName + ' ' + @FirstName) end end) 
    return @Result 
end 


create table Person
(
  isperson bit,
  lastname nvarchar(100),
  firstname nvarchar(100),
  fullname as [dbo].[FullNameLastFirst] (isperson, lastname, firstname)
)
go
insert into person(isperson, lastname, firstname) values (1,'Firstname', 'Surname')
go

create index ix1_person on person(fullname)
go

select fullname from Person with (index=ix1_person) where fullname = 'Firstname Surname'
go