Dattaprabhakar G. answered • 09/08/14
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Ken:
Look at the change in y = f(x) as x changes successively. Determine the slope at x = Change in y/ Change in x, at x
Change in x, Change in y = f(x) Slope
+1 -3 -3
+2 -6 -3
+2 -6 -3
Since the slope is the same, the function f(x) is linear on [3.8]. Who knows what is its shape outside this interval.
Dr. G.
Dattaprabhakar G.
Your are correct, Phillip R. But that is not consistent with the question, "(a) Find a formula for the function". Actually, you have done it for Ken.. What I wanted him to realize, is that he should NOT think that if the function is linear on [3,8], it is linear everywhere. The table does not give us any information about the shape of the function outside [3, 8].
One COULD definitely say that "it (the function) is unknown to be linear within the interval because these are merely points on the graph of the function but we have no information about the behavior of the function anywhere but x = 3, 4, 6 and 8". That would be too extreme and too pessimistic an approach.
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09/08/14
Dattaprabhakar G.
Phillip R. Your comment raises an interesting question. What subset of [3, 5] is enough,so that a if a linear function is defined on it, that will extend to the whole interval? For example, if we say that f(x) is linear on the set of rationales in [3, 5], would that mean f(x) is linear on [3, 5]? I do not know the answer to this question. But some mathematicians would certainly know. Perhaps there is a "well known" extension theorem. It is too abstract for me. I like Statistics.
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09/08/14
Phillip R.
09/08/14