The Gaussian Function $y = e^{-x^2}$

$y = e^{-x^2}$ is obtained by placing $-x^2$ in the exponent of the exponential function $e^u$, producing the famous symmetric bell curve. Because it is the kernel of the normal distribution, it appears throughout probability, statistics, physics, and signal processing.

Domain and range The function is defined for every real number $x$. Since $-x^2 \le 0$, we have $0 < e^{-x^2} \le 1$, so the range is $(0, 1]$. The value equals $1$ only at $x = 0$ and stays strictly positive everywhere else.

Symmetry Because $f(-x) = e^{-(-x)^2} = e^{-x^2} = f(x)$, the function is even and its graph is symmetric about the $y$-axis.

Monotonicity and extrema The derivative is $f'(x) = -2x\,e^{-x^2}$. It is positive for $x < 0$ and negative for $x > 0$, so the curve increases up to the peak $(0, 1)$ and decreases afterward. This maximum is the only extremum; there is no minimum.

Asymptotes and limits As $x \to \pm\infty$ the exponent tends to $-\infty$, so $e^{-x^2} \to 0$ and the $x$-axis ($y = 0$) is a horizontal asymptote. The decay is extremely fast; for example $f(3) \approx 0.000123$.

Inflection points The second derivative is $f''(x) = (4x^2 - 2)\,e^{-x^2}$, which vanishes at $x = \pm \dfrac{1}{\sqrt{2}} \approx \pm 0.707$. There the curve changes from concave down to concave up, with $y = e^{-1/2} \approx 0.607$. These are the "shoulders" of the bell.

Relation to other functions The standard normal density $\dfrac{1}{\sqrt{2\pi}}\,e^{-x^2/2}$ is just this function rescaled horizontally and normalized so its total area is $1$. The area under the basic curve is the celebrated Gaussian integral $\displaystyle\int_{-\infty}^{\infty} e^{-x^2}\,dx = \sqrt{\pi}$.

Applications and history Through the work of de Moivre, Gauss, and Laplace in the 18th and 19th centuries, this curve became central to the theory of errors and probability. Today it models measurement error, serves as the fundamental solution of the heat equation, blurs images as the Gaussian filter, and underlies kernel methods in machine learning.