Answer
$T(-3,2,5)=(0,-1,7)$
The pre-image is $\{(2-y,y,5-y):\text{y is real}\}$
(Other parametrizations are possible.)
Work Step by Step
We know that $T(v_1,v_2,v_3)=(0,v_1+v_2,v_2+v_3)$
Thus $T(-3,2,5)=(0,-3+2,2+5)=(0,-1,7)$
Let $(x,y,z)$ be the pre-image of $(0,2,5)$. Then $T(x,y,z)=(0,x+y,y+z)=(0,2,5)$.
Thus $x+y=2\\x=2-y$ $\quad$ $y+z=5\\z=5-y$.
From this we get
$x=2-y,y=y,z=5-y.$
So the pre-image is $\{(2-y,y,5-y):\text{y is real}\}$
