The Witch of Agnesi $y = \dfrac{1}{1+x^2}$

$y = \dfrac{1}{1+x^2}$ is a rational function known as the Witch of Agnesi, and its graph is a bell shape closely resembling the Gaussian. Because its denominator is a polynomial rather than an exponential, its tails fall off more gently than the Gaussian's.

Domain and range The denominator $1 + x^2$ is always at least $1$ and never zero, so the function is defined for all real $x$. Since $0 < \dfrac{1}{1+x^2} \le 1$, the range is $(0, 1]$.

Symmetry Because $f(-x) = \dfrac{1}{1+(-x)^2} = f(x)$, it is an even function, symmetric about the $y$-axis.

Monotonicity and extrema The derivative is $f'(x) = \dfrac{-2x}{(1+x^2)^2}$. It is positive for $x < 0$ and negative for $x > 0$, so the curve reaches its maximum at the peak $(0, 1)$ and decreases on either side.

Asymptotes and limits As $x \to \pm\infty$ the denominator grows without bound and $y \to 0$, so the $x$-axis is a horizontal asymptote. The decay is slow, of order $1/x^2$; for example $y = 0.1$ already at $x = 3$.

Inflection points The second derivative $f''(x) = \dfrac{6x^2 - 2}{(1+x^2)^3}$ changes sign at $x = \pm \dfrac{1}{\sqrt{3}} \approx \pm 0.577$, where $y = \dfrac{3}{4}$. These shoulders mark the change from concave down to concave up.

Relation to other functions This function is exactly the derivative of the arctangent: $\dfrac{d}{dx}\arctan x = \dfrac{1}{1+x^2}$. Its total integral is $\displaystyle\int_{-\infty}^{\infty}\dfrac{dx}{1+x^2} = \pi$, and dividing by $\pi$ gives $\dfrac{1}{\pi(1+x^2)}$, the probability density of the Cauchy distribution.

Applications and history The curve is named after the 18th-century Italian mathematician Maria Gaetana Agnesi (1718–1799). When it appeared in her 1748 textbook, the Italian word "versiera" (a turning curve) was confused with "avversiera" (she-devil), leading to the famous English mistranslation "witch." Today it arises as the Cauchy distribution in probability and the Lorentzian function describing resonance.