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!
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.