How to sketch output PDF given the transformation?
There is no impulse because there is zero probability mass at $0$. As the problem itself shows, it is perfectly all right for a pdf to make sudden jumps in value; note that the graph of $f_X$ (yes, I do mean $X$, not  $Y$) jumps in value at $0$ and the value of $f_X(0)$ cannot be determined from the graph. $f_X(0)$ could have value $\frac 14$ or $\frac 12$ or any value in between (or any nonnegative value for that matter)