Question

The point (1, 1) is the image under the translation(x,y)->(x+3, y-3) . What is the preimage of this point?

Answer 1

Let us call the translation f( x,y )

The preimage of a point (x,y) under f is composed of all points (a,b) sich that (x,y) = f( a,b).

That means for this specific f we have
x = a+3
y = b -3

We are given the point (x,y) = (1,1), which we can plug into our two equations to get

1 = a + 3 hence a = -2
1 = b - 3 hence b = 4.

That means the preimage of (1,1) under the given translation consists only of the point (-2,4).

<3