pdf of inverted random variable

Question

Suppose i have an uniform distributed random variable $X$ on $[5,30]$. I would like to find pmf of $Y = \frac{1}{X}.$

\begin{align*} Y =& \frac{1}{X}\\ dY =& -\frac{1}{X^2}dX\\ dY =& -Y^2dX\\ -\frac{1}{Y^2}dY =& dX\\ \end{align*}

I believe that $$f_{X}(x) = \frac{1}{25}dX$$ so $$f_{Y}(y) = -\frac{1}{25}\frac{1}{Y^2}dY$$ But lecture notes give me answer without minus sign. Where i made a mistake?

Can it be because we kinda swap range borders because of division and how to express it formally?

Answer 1

$$f_X(x)=\frac{1}{25}$$ $$f_Y(y)=f_X(x)\bigg|\frac{dX}{dY}\bigg|=\frac{1}{25}\frac{1}{y^2}$$