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i do not see the domain changing it is just the variable changing from t TO t-3

Edit: *f(u) has a domain of 2 to 4.

i stick with basking

Good point, basking!
And Stacey! Thanks for your reply!
I considered the domain of f(t) to be limited when posted my question! And I guess that's the first thing that comes in everybody's mind when seeing such question in math! Otherwise, I'd mention that.
I took a glance at that book again, and found an example of that section concerning such topics. I'll attach a shot of a related graph from the book to this post.
But about Example 1, I ought to say that that's a practical situation, while we're talking about a problem in pure math. In my book, the Example 2 works in here. That's what I'm talking about.

the two functions are the same

Again we have ambiguity. f does not change. But f(t-3) is not the same function of t as f(t).

THE FUNCTIONS BEHAVE SIMILLARLY