Fort Worth claims to be 'Where the West Begins' because many years ago someone in the US Geo Service was tasked with determining where the 'west' was. They picked a longitude line, said that was it, and the line ran right through Fort Worth. Or at least close to it.

In Oklahoma this line is known as the 'Indian Meridian' and is 97d, 14m, 33sec. degrees Long. Fort Worth is also on this Longitude.

And everything west of that is west.

But to be practical you really don't start feeling the southwest until you get at least to San Saba or Sweetwater.