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I often see the following inequality is used over and over again $ 1−x⩽e^{−x} $ for $x \in \mathbb{R}$, for proving or deriving various statements.

As a layman, I haven't seen this inequality appearing in any class I have taken in my life. So it seems quite unnatural to me, and seems just a special result for linear function and exponential function. It is not yet part of my instinct to use it for solving problems. So I want to fill up this indescribable gap within my knowledge.

I wonder if there are other similar results for possibly other commonly seen functions (elementary functions?).

Is there some source listing such results?

Thanks and regards!

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    As I said: It is reasonable to expect you to understand things that have never been explicitly covered in courses, since many things are covered only implicitly. ncmathsadist's answer pretty much covers the reasons why one would expect this inequality to hold.2012-07-02

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This inequality occurs for two reasons. The line $y = 1-x$ is tangent to the curve $y=e^{-x}$ at $(0,1)$ and $x\mapsto e^{-x}$ is concave up everywhere. Hence the line lies below the curve.

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    I have taught calculus for many years and the common parlance there is "concave up" or "concave down". It's an ingrained habit.2012-07-02