# Cumulative Distribution Function with New Random Variable

I'm really new to statistics and probability so sorry if this is a really basic question. I just wasn't sure how to do it. I tried looking it up but can't find much information.

If I'm given a cdf for a random variable X, how do I find it for a new random variable Y which was in terms of X? Do I just plug in Y? My example is following:

FX(x) = 1 −1/x
for 1 < x < ∞
Find the cdf for the new random variable Y = -X + 2


\begin{align} F_Y(y) & = \Pr(Y\le y) = \Pr(-X+2\le y) = \Pr(X\ge 2-y) \\[8pt] & = 1-\Pr(X\le 2-y) = F_X(2-y) \end{align} provided $2-y\ge 1$, which is the same as $y\le 1$.
If $y>1$ then $$F_Y(y) = \Pr(Y\le y) = \Pr(-X+2\le y) = \Pr(X\ge 2-y) = 1.$$