$\tan(x)$ outputs go from $-\infty$ to $+\infty$ with each step equals $\pi$
My question is how to modify or build a function based on $\tan(x)$ that outputs of the function will change from $-\infty$ to $+\infty$ with a step of $1$ instead of $\pi$?
You noticed that those individual curves between the asymptotes (where y = tan(x) approaches $\infty$ from one side a $-\infty$ from the other) each cover a horizontal distance of $\pi$.
This is called the period of the function. $y = \tan x$ has a period of $\pi$. (And $y = \sin x$ and $y = \cos x$ both have a period of $2\pi$.)
In general, if you have periodic function $y = f(x)$ with a period $p$, the function $y = f(Bx)$ will have a period of $\omega = \frac p B$.
Since $y = \tan(x)$ has a period of $\pi$, to find the value of $B$ such that $y = \tan(Bx)$ has a period of 1, take the formula $\omega = \frac p B$ and plug in 1 for the desired period $\omega$, a $\pi$ for the period of $y = \tan(x)$ and solve for $B$.
$1 = \frac \pi B$ will pretty clearly give you $pi$ for B.
Therefore, $y = \tan (\pi x)$ will have a period of 1.