How Much Should Scientists Extrapolate The Known To The Unknown?

We know an awful lot about the observable universe in the "low-energy" regime and a question that bugs my mind is: how much should we extrapolate what we know about the universe to regions we technically don't know much about? Three recent events have made me think about this:

1. Dr. Kolb and Turner, who probably gave the most enjoyable talks at the AAS Meeting, bristled at the idea that many cosmologists are convinced we live in a multiverse from extrapolating what we know about the observable universe and went into outright mockery of string theorists who have dreamed up branes, landscapes, etc... by doing the same thing.
2. Sean Carrol, who recently polled readers of how likely inflation was, suggested one of the reasons he only gives inflation a 75% chance of being real is that it involves energy scales that we know nothing about. (But in fairness to inflation, this leads to predictions that are verified so you can't put in in the same category as something like string theory. :))
3. Jonathan recently asked if it was fair to say the universe is flat globally if we only have data showing flatness "locally".  

Now, first of all I think there is one very good reason to extrapolate: It's the best we can do given what we know now!

But I guess one question that bothers me is: How much confidence should we put in our extrapolation of known physics to the unknown?

I will give you an overly simplistic example just to make sure we are on the same page.  What if we were living in a one dimensional world and this was our observable universe:

Now, what would we, living at that red dot, conclude about the whole universe from extrapolation?  Probably that we live on a sine wave.  And from x between -6 and 6 that assumption would seem to work very well.

But what if it turns out this was our universe:

As we see here, our assumptions based on extrapolation would be wrong.

Now many of you will think this is childish, but is it?

For example: there was a day when humans had good reasons to believe the earth was flat.  They extrapolated... and they were wrong.  There were others who had good evidence that the sun and stars orbited around the earth.  They extrapolated... and they were wrong.  There were others who had experimental reasons to think gravity can effect all other matter instantaneously. (Action at a distance)  They extrapolated... and they were wrong. Etc...

And so I wonder how likely we are to be falling into the same trap as our ancestors?  On one hand extrapolation is all, and is the most honest thing, we can do.  But on the other, history has had a way of punishing the extrapolaters.  

One should be said though: once new data does come demonstrating our cherished "extrapolated" theories are wrong, eventually the mainstream science community drops them and moves on.  History has shown that eventually mainstream science will lead toward the right answer as more data comes in on the matter.

So what do you think?  How much confidence should we scientists put in our "theories from extrapolation"?