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 $x < 0$.

A tug-of-war between growth and decay

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 $0$.

Monotonicity and maximum

By the product rule, the derivative is as follows.

Since $e^{-x} > 0$, the sign is governed by $1 - x$: the function increases for $x < 1$ and decreases for $x > 1$. It therefore has a maximum at $x = 1$, of value $e^{-1} = \dfrac{1}{e} \approx 0.368$.

Asymptote and inflection point

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.

Past the hump the curve tapers gently toward $0$.

An asymmetric shape

The curve is not symmetric: the left side of the hump rises abruptly from the origin, while the right side trails off in a long tail. The area under the part with $x \geq 0$ is as follows.

This is the value $\Gamma(2) = 1! = 1$ of the gamma function, and it shows that $x^k e^{-x}$ normalizes neatly into a probability density.

Applications

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