Domain and range of this function?

• Sep 2nd 2011, 05:40 PM
explodingtoenails
Domain and range of this function?
Find the domain and range of this function:

f(x) = cosx/x.

I just took a quiz today and I blanked out. I wasn't sure what notation you are supposed to use for this kind of problem so I probably got it wrong.
• Sep 2nd 2011, 07:20 PM
Prove It
Re: Domain and range of this function?
Quote:

Originally Posted by explodingtoenails
f(x) = cosx/x

I just took a quiz today and I blanked out. I wasn't sure what notation you are supposed to use for this kind of problem so I probably got it wrong.

You should know that a rational function with numerator and denominator both defined over the reals is also defined over the reals EXCEPT where the denominator is 0.

So what is the domain of this function?
• Sep 2nd 2011, 08:10 PM
explodingtoenails
Re: Domain and range of this function?
The domain would be (-infinity, 0) U (0, infinity)?
• Sep 2nd 2011, 08:17 PM
Prove It
Re: Domain and range of this function?
Quote:

Originally Posted by explodingtoenails
The domain would be (-infinity, 0) U (0, infinity)?

Correct, though it's easier to write $\displaystyle \mathbf{R}\backslash\{0\}$

Now the range is a bit tougher, but not impossible. You should know that the cosine function oscillates between -1 and 1. What happens when the denominator is large? When the denominator is small?
• Sep 2nd 2011, 08:56 PM
explodingtoenails
Re: Domain and range of this function?
When the denominator is large, the number is smaller, and vice versa when the denominator is small.

Would the range be (-infinity, +infinity)?

Also, how do I know which notation to use when doing domain and range of a function? ie. The parenthetical notation or using "less than or equal to" signs?
• Sep 2nd 2011, 09:36 PM
Prove It
Re: Domain and range of this function?
Quote:

Originally Posted by explodingtoenails
When the denominator is large, the number is smaller, and vice versa when the denominator is small.

Would the range be (-infinity, +infinity)?

Also, how do I know which notation to use when doing domain and range of a function? ie. The parenthetical notation or using "less than or equal to" signs?

Yes, because for small negative values of x, the function shoots to $\displaystyle -\infty$, while for small positive values of $\displaystyle x$ the function shoots to $\displaystyle +\infty$. Of course, it's easier to write $\displaystyle \mathbf{R}$ instead of $\displaystyle (-\infty, \infty)$.

All notations are equivalent, it's just personal preference (I go for whichever one is easiest to write in each case).
• Sep 3rd 2011, 12:41 PM
explodingtoenails
Re: Domain and range of this function?
Just as an extra question, would the limit as x approaches infinity of cosx/x be infinity?
• Sep 3rd 2011, 12:52 PM
Prove It
Re: Domain and range of this function?
Quote:

Originally Posted by explodingtoenails
Just as an extra question, would the limit as x approaches infinity of cosx/x be infinity?

No, since $\displaystyle \cos{x}$ is bounded, that means as the denominator (x) gets extremely large, the fraction gets extremely small. So the limit would be 0.
• Sep 3rd 2011, 01:08 PM
explodingtoenails
Re: Domain and range of this function?
Quote:

Originally Posted by Prove It
No, since $\displaystyle \cos{x}$ is bounded, that means as the denominator (x) gets extremely large, the fraction gets extremely small. So the limit would be 0.

Then how come when I graph it, it fluctuates between positive and negative values when the x-value gets larger?
• Sep 3rd 2011, 01:12 PM
Prove It
Re: Domain and range of this function?
Quote:

Originally Posted by explodingtoenails
Then how come when I graph it, it fluctuates between positive and negative values when the x-value gets larger?

I suggest you zoom out so that you get a better picture of what the function's behaviour is. It fluctuates because the cosine function fluctuates. But you should notice that they fluctuate less and less as the x values increase. Even though the function will continue to oscillate forever, as the denominator increases, the function will still get infinitesimally small so that the function is indistinguishable from 0.
• Sep 3rd 2011, 01:20 PM
explodingtoenails
Re: Domain and range of this function?
Quote:

Originally Posted by Prove It
I suggest you zoom out so that you get a better picture of what the function's behaviour is. It fluctuates because the cosine function fluctuates. But you should notice that they fluctuate less and less as the x values increase. Even though the function will continue to oscillate forever, as the denominator increases, the function will still get infinitesimally small so that the function is indistinguishable from 0.

Ahh dang it. That means I did quite badly on my AP Calc quiz that I took yesterday. :(