In programming, often the value of True is also 1, and False is 0.
This means that:
(x>5)*4will return 4 if x is greater than 5 (because (x>5)==1), else 0.
I need to accomplish a similar thing using mathematical operators (no piecewise functions, this has to be typed into Desmos calculator.)
Specifically, I need $$ f(x)=\begin{cases} 1&&\text{if}~ x\leq n \\ 0&&\text{otherwise} \end{cases} $$ without having to use piecewise notation. Here $n$ is a positive integer, as is $x$.
$\endgroup$ 43 Answers
$\begingroup$(Posted by Dave Radcliffe but then deleted):
$\endgroup$ 2 $\begingroup$The Desmos calculator supports piecewise functions such as
{x<5 : 0, 1}or{x<=5 : 0, 1}
$$ x\mapsto \frac{|2n+1-2x| + (2n+1-2x)}{2|2n+1-2x|} = \begin{cases} 1 & \text{if } x\le n, \\ 0 & \text{if } x>n. \end{cases} $$
$\endgroup$ $\begingroup$To simplify Hardy's answer, a function I commonly use is
$$\frac{1}{2} + \frac{n-x}{2|n-x|}$$
If you want to make calculations less taxing by working only with integers during calculations, this is obviously the same as
$$\frac{1+\frac{n-x}{|n-x|}}{2}$$
This solution immediately arises from the fact that $\frac{n-x}{|n-x|}$ is either $1$ when $x<n$ and $-1$ when $x>n$
This is obviously undefined for equality of the two variables, although you can add a factor to the top and bottom to change the undefined point, such as
$$\frac{1}{2} + \frac{n-x+1}{2|n-x+1|}$$
which is undefined not at equality, but when $x=n+1$
This is not needed however, as Desmos has native support for piecewise functions... see this link for an explanation by the Desmos devs
