Application of Convolution Formula

Question

I got the following from my statistics lecture today.

Can someone explain to me the last 2 steps?
Specifically, how $f_Y(t-x)=1$

Thanks!

Edit 1:

Answer 1

Note that $f_Y(y)$ is the pdf of the uniform distribution, hence, $$ f_Y(y) = \mathbb{I}_{(0 \le y \le 1)} $$ and in your case you are looking at $f_Y(t-x)$, which is $1$ whenever $0 \le t-x \le 1$ and zero otherwise.

The first case inequality implies $x \le t \le 1+x$.