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Filling in the blank to a statement

  1. Oct 14, 2008 #1
    1. The problem statement, all variables and given/known data
    For the statement "If x is within _______ units of 3 (but not equal to 3), then f(x) is within 0.01 unit of 2," write the largest number that can go in the blank

    2. Relevant equations

    3. The attempt at a solution
    It's a multi-step problem, but the two numbers I ended up with are 1.99 or 2.01. I'm not sure which one fits into this statement because I'm not quite sure what it's asking.
  2. jcsd
  3. Oct 14, 2008 #2


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    This question doesn't make sense unless you are given a specific function f! It looks to me like f is some continuous function such that f(3)= 2. But what goes in the blank depends upon exactly what f is. Surely you realized that?
  4. Oct 14, 2008 #3
    Sorry, maybe I should have put the function that I was given. I'm not sure what F! means, but here's the function I'm given. F(x) = (x^3 - 7x^2 + 17x - 15)/( x - 3) It is a continuous graph, but at x = 3 there's a point of discontinuity which is (3,2).
  5. Oct 14, 2008 #4


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    The exclamation point after f was punctuation in the sentence, just like this one!
    The graph can't be continuous AND have a point of discontinuity. Without looking at the graph, I'm guessing that there is a "hole" at (3, 2). I'm also guessing that if you divided the numerator polynomial (x^3 + ...) by (x - 3) there wouldn't be a remainder. That might be a hint.
  6. Oct 15, 2008 #5


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    You mean that F has a removable discontinuity at x= 3. That's a discontinuity because 33- 7(32+ 17(3)- 15= 27-63+ 51- 15= 78- 78= 0 as well as 3- 3= 0: both numerator and denominator are 0 at x= 3. Because x= 3 makes the numerator 0, we know that x-3 is a factor. Knowing that it is easy to see that x3- 7x2+ 17x- 15= (x-3)(x2- 4x+ 5) so for x NOT equal to 3, this is just F(x)= x2- 4x+ 5= (x-4)(x-1). The discontinuity is "removable" because that has limit 3 at x= 2.

    Now, if "f(x) is within 0.01 unit of 2", that is, if [itex]1.99\le x^2- 4x+ 5\le 2.01[/itex] what must x be?
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