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Functions question



I have no idea what to do about this question as at x=2, the function is undefined because we're not allowed to divide by zero. Wikipedia says that a continuous function is "In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output". That isn't helping me at all.

Any help/hints would be appreciated.
(edited 11 years ago)
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Avatar for sputum
sputum
9
Original post by Zishi


I have no idea what to do about this question as at x=2, the function is undefined because we're not allowed to divide by zero. Wikipedia says that a continuous function is "In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output". That isn't helping me at all.

Any help/hints would be appreciated.


The idea is that you use the function for the domain (which excludes x=2) and try to 'stitch it together' in some sense by just defining what your function is at x=2 if you can.
ie can you make this function continuous if you define a value for your function at x=2?

EDIT: or imagine graphing it. What does it do near x=2?
(edited 11 years ago)
Reply 2
Avatar for SalBhatti
SalBhatti
OP
13
Original post by sputum
The idea is that you use the function for the domain (which excludes x=2) and try to 'stitch it together' in some sense by just defining what your function is at x=2 if you can.
ie can you make this function continuous if you define a value for your function at x=2?

EDIT: or imagine graphing it. What does it do near x=2?


As I put values of x to be -1, 0, and 1, I see that values of f(x) increase in negativity, i.e -1.66...,-2.5,-5. So does that mean that value of f(x) as x=2 should be more negative than -5?
Do you know what the graph of f(x)=1x f(x) = \frac{1}{x} looks like?

If so, then then try to sketch f(x)=1x−2 f(x) = \frac{1}{x - 2} and then f(x)=5x−2 f(x) = \frac{5}{x - 2} , by using standard graph sketching rules.

If not, then what happens when x gets really close to 2 from above? Take x = 2.1, 2.05, 2.01 and see what happens to 5x−2 \frac{5}{x - 2} . Then try taking x = 1.9, 1.95, 1.99, and see also what happens.

A continuous function has a precise mathematical definition, but intuitively, it means that you can draw it without taking your pen off the page. Note that f(x)=x f(x) = x is continuous, as you can easily draw a straight line without ever taking your pen from the page. Do you see what happens to the graph in this case?
(edited 11 years ago)
Original post by Zishi


I have no idea what to do about this question as at x=2, the function is undefined because we're not allowed to divide by zero. Wikipedia says that a continuous function is "In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output". That isn't helping me at all.

Any help/hints would be appreciated.


A function can appear undefined at certain values for which it actually can be defined. The standard way of checking this is by evaluating the limit of the function when the variable approaches the indeterminate value.

Classic example:

Spoiler



In your case, evaluate lim⁡x→2±5x−2\displaystyle \lim_{x\to 2^{\pm}} \dfrac{5}{x-2} (think of the curve) and see whether the value(s) you get are definite or not. If so, compare your result to the possible answers. :smile: A function is continuous if (basically) you can sketch it without taking your pen off the page... Beware though, a function that is defined over a certain domain does not imply it is continuous over that domain, but I don't think you have to worry about that here.
(edited 11 years ago)
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Avatar for SalBhatti
SalBhatti
OP
13
Oh, thanks a lot everyone. I actually didn't think that it's a limits question! :h:

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