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

$y = x e^{-x}$ is the product of the linear term $x$ and the decaying exponential $e^{-x}$. Its domain is all real numbers and it passes through the origin $(0, 0)$; it is positive for $x > 0$ and negative for $x < 0$.

The factor $x$ tries to grow with $x$, while $e^{-x}$ tries to push the value toward $0$. The result of this tug-of-war is the characteristic shape that rises to a hump and then approaches $0$. By the product rule, the derivative is

Since $e^{-x} > 0$, the sign is governed by $1 - x$: increasing for $x < 1$ and decreasing for $x > 1$. Hence there is a maximum at $x = 1$ with value $1 \cdot e^{-1} = \dfrac{1}{e} \approx 0.368$.

As $x \to +\infty$, the exponential decay beats the linear growth, so the function approaches $0$ and the $x$-axis is a horizontal asymptote. As $x \to -\infty$, with $x < 0$ and $e^{-x} \to +\infty$, it diverges to $-\infty$. The second derivative $y'' = (x - 2)e^{-x}$ changes sign at $x = 2$, which is therefore an inflection point (value $2e^{-2} \approx 0.271$).

Representative values are listed below; beyond the hump it tapers gently toward $0$.

• $x = 0 \Rightarrow y = 0$
• $x = 1 \Rightarrow y = \dfrac{1}{e} \approx 0.368$ (maximum)
• $x = 2 \Rightarrow y = \dfrac{2}{e^2} \approx 0.271$ (inflection)
• $x = 3 \Rightarrow y = \dfrac{3}{e^3} \approx 0.149$

This is the simplest example ($k = 1$) of the form $x^k e^{-x}$ and shares its skeleton with the probability densities of the gamma and Erlang distributions. It is widely used to model phenomena where a growing factor competes with a decaying one, as in queueing theory, signal processing, and chains of radioactive decay.