Which of the following are monotonic transformations?

[CaitlinH86]

I am really struggling trying to grasp the philospohy of montonic transformation. The question is: Which of the following are monotonic transformations?

(a) k x u[x1, x2] where k is a real positive number (I think Yes?)

(b) -2 x u[x1, x2] (I think No because of the -2, but that's just a guess.)

(c) Square root of u[x1, x2] (NO IDEA)

(d) 1/u[x1, x2] (NO IDEA)

I was reading other posts on the same type of problems and I guess I'm also confused on how you would even graph this. I understand that if the graph goes negative and positive then its not a monotonic transformation, but how do I know that for the above questions? Please help!

 

[talk2glenn]

A monotone transformation preserves the behavior of a function. So if u(x1, x2) is monontonically increasing on a given interval, then its behavior must be preserved in k*u(x1, x2).

In -2*u(x1, x2), the behavior of the transformed function varies in direction by the sign of u. For example, assume for any value of x1, u(x1) is strictly increasing. The transformed function does not preserve this behavior, and is not a monotonic transformation. Note that what the functions "actually do" is irrelevant; they aren't graphable, but you don't need to graph them to answer the question. They are general case questions. If it helps, you can fill in arbitrary cases to test hypothesis about functional behavior; if you find one case where order is violated, you've disproved monotonicity.

Looking at the third one, we have sqrt[u(x1, x2)]. Again, assume u is increasing for any value of x1. Does taking the sqrt of the function change this behavior? Certainly slope behavior may be altered (eg, consider u as a linear function), but order is preserved. This is a monotonic transformation.

Try to figure d out for yourself, and prepare yourself for a warning on the use of homework forums :)

 

[CaitlinH86]

Ok, thanks. That definitely helps. And what sort of warning? I did the work myself, I just want to grasp the concepts better.