# Filling in the blank to a statement

1. Oct 14, 2008

### MellowOne

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
Nonr

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. Oct 14, 2008

### HallsofIvy

Staff Emeritus
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?

3. Oct 14, 2008

### MellowOne

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).

4. Oct 14, 2008

### Staff: Mentor

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.

5. Oct 15, 2008

### HallsofIvy

Staff Emeritus
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 $1.99\le x^2- 4x+ 5\le 2.01$ what must x be?