As the title says, I need to create/generate a formula for a function with natural domain $(-\infty ,0] \cup [10, \infty )$ and with range $(-\infty , -1]$
where $-\infty $ means going off towards negative numbers' infinite, and $\infty$ means reaching towards positive infinite. $\cup$ means union, so all numbers from both sets $(-\infty ,0] \cup [10, \infty)$ combined. it means elements from $(-\infty ,0]$ "or" $[10, \infty)$
Also known as all numbers except for $(0,10)$ or $[1,9]$
how do you generate this type of function with a center section of the domain missing? I am able to create a function where there is only one type of restriction, but not a function where there's a set of numbers restricted.
Best Answer
We will change the problem a bit, to natural domain $(-\infty,-5]\cup [5,\infty)$. If we solve that problem we can shift to get what we want. We are doing this because symmetry is nice.
And although I have nothing really against $5$, but $1$ is a nicer number. So we will deal with natural domain $(-\infty,-1]\cup [1,\infty)$. You can do the suitable scaling.
So we want to run into trouble in $(-1,1)$. The following sounds good: $$\sqrt{1-\frac{1}{x^2}}.$$
But the range is $[0,\infty)$. That part is easy to fix: switch sign, and subtract $1$.