Finding an image under a given transformation

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SUMMARY

The discussion centers on the mathematical transformation of a graph defined by the line y = b within the bounds of -a ≤ x ≤ a. The participant concludes that setting v = 1 is invalid because it represents a single point rather than a line segment. The graph is constrained by the rectangle defined by |u|, |v| ≤ 1, but it does not fully occupy this rectangle; instead, it is contained within the image of another rectangle defined by |x| ≤ a, |y| ≤ b.

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Miike012
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My question is in the paint document.
And I think I know the answer to my question. I asked why can't I let v = 1 then my first first region transformation would

the line y = b between -a≤x≤a.

The reason I think I can't do this is because the end point v = 1 is a point and not a line segment...
 

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The graph is bounded by the rectangle ##|u|,|v|\le 1## but it isn't that rectangle. It is just contained in it. And the other graph is contained in the image of that rectangle, which is ##|x|\le a,|y|\le b##.
 

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