Determine if the function is a contraction! Thank you!

**jack56890** · February 2nd 2013, 09:57 AM

The function is g(x) = (1/2)*(e^(-x)) x is R

Thank you!!!!

**emakarov** · February 2nd 2013, 10:12 AM

Use the mean value theorem.

**hollywood** · February 2nd 2013, 06:38 PM

What happens as x goes to $-\infty$ ? For it to be a contraction, $|g'(x)|$ has to be less than 1, right?

- Hollywood

**HallsofIvy** · February 3rd 2013, 11:41 AM

Actually, there are several different ways to define "contraction". jack56890, what definition are you using?

**hollywood** · February 3rd 2013, 09:01 PM

My definition is that a function is a contraction if there is a real number c, $0\le{c}<1$ such that $|g(x)-g(y)|\le{c}|x-y|$ for all $x,y\in\mathbb{R}$ where $g(x)$ and $g(y)$ are defined. Just so that you can put my post in context - if you're using a different definition, my post might not make sense.

- Hollywood

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