question on transformation

Question

If a $2$d coordinate transformation function is given by $f(x,y)= 3x+1$, then what does it mean? How do I calculate the transformed coordinates for the points say $(3,4)$ in the initial space?

Answer 1

If you want to take points on $\mathbb{R}^2$ and map them or transform them to $f(x,y)=3x+7y$, this doesn't really define a transformation, without more addtional information. Essentially from what I've gathered, you just want to map the set of all points in $\mathbb{R}^2\to f(x,y)$. This just defines a multivariable function.

So you're in a sense transforming the plane of all pairs of points in the $xy$-plane to the function you defined. If you want to see what happens at a given point, just plug in the $x$ and $y$ values.

For example $(2,3)$:

$f(2,3)=3(2)+7(3)=6+21=27$.

But also if you just want $f(x,y)=3x+1$ this defines an infinite number of lines, since it does not depend on $y$. You can graph these lines on paper and see what happens.

Answer 2

If $f(x,y) \equiv 3x+1$, then $f(3, 4) = f(3, y) = 10$. $f(x,y)$ is not dependent to y. The $10$ value is also for $f(3, 0) = f(3, -10) = f(3, 2e45) = \cdots = 10$ and so on.