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.
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.
Correct, though it's easier to write $\displaystyle \displaystyle \mathbf{R}\backslash\{0\}$
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?
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 $\displaystyle \displaystyle -\infty$, while for small positive values of $\displaystyle \displaystyle x$ the function shoots to $\displaystyle \displaystyle +\infty$. Of course, it's easier to write $\displaystyle \displaystyle \mathbf{R}$ instead of $\displaystyle \displaystyle (-\infty, \infty)$.
All notations are equivalent, it's just personal preference (I go for whichever one is easiest to write in each case).
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.