Do convex lenses start to act like concave lenses at twice their focal length?

Question

I'm trying to create a cheap concave lens effect for a class I'm doing. It seems like a convex lens starts to create a similar effect anywhere passed twice it's focal length. It also makes everything upside down, but that's okay.

Are the effects of a convex lens a demonstrational equivalent to a concave lens once you pass double the focal length? Ignoring the upside down aspect.

This is for a hand made kaleidoscopes library program to give you some idea of the stakes.

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EDIT: I set up a magnifying glass and walked some distance away from it. What I see through it is:

1. upside down,
2. in focus, and
3. inclusive of a wider view than what it blocks. I can see an entire shed within the lens, while it does not block me from seeing the shed around it.

Answer 1

No. The light has still "converged" with respect to the horizontal angle. Draw arrows on the ends of the lines to understand. I think the light rays appear to be diverging to you because they are moving away from eachother but what matters is the way the angle changes before and after the lens. It's just that all the lines are crossing the horizontal axis, you're just seeing the continuation of those lines.

The only thing that happens after the point at which the light focuses is that it becomes unfocused.