Graph of the Exponential Decay $y = e^{-x}$

$y = e^{-x}$ is a decaying exponential with base $e \approx 2.718$, and can be written $e^{-x} = \dfrac{1}{e^x} = \left(\dfrac{1}{e}\right)^x$. Its domain is all real numbers and its range is $y > 0$; the value is never zero or negative.

The derivative $y' = -e^{-x}$ is always negative, so the function is monotonically decreasing everywhere. Each time $x$ increases by $1$, the value is multiplied by $\dfrac{1}{e}$ (about $0.368$).

As $x \to +\infty$ it approaches $0$, so the $x$-axis ($y = 0$) is a horizontal asymptote; as $x \to -\infty$ it diverges to $+\infty$. It always passes through $(0, 1)$. The second derivative $y'' = e^{-x} > 0$, so the graph is always convex (concave up).

Concretely, $y = 1$ at $x = 0$, $y = \dfrac{1}{e} \approx 0.368$ at $x = 1$, $y = \dfrac{1}{e^2} \approx 0.135$ at $x = 2$, and $y = e \approx 2.718$ at $x = -1$.

It is the mirror image of $y = e^x$ across the $y$-axis: where $e^x$ grows explosively, $e^{-x}$ decays just as fast. It is closely tied to $\ln x$ as well: setting $e^{-x} = t$ gives $x = -\ln t$.

Decaying exponentials appear throughout nature. Radioactive decay, Newton's law of cooling, the discharge of a capacitor, and the elimination of a drug from the body — any process that decreases in proportion to the amount currently present takes the form $e^{-kx}$. It also underlies the exponential distribution in probability.