The Rational Function $y = \dfrac{x}{1+x^2}$

$y = \dfrac{x}{1+x^2}$ is a rational function with numerator $x$ and denominator $1+x^2$. It can be viewed as the Witch of Agnesi $\dfrac{1}{1+x^2}$ multiplied by $x$, and its defining feature is a single upward bump (crest) and a single downward bump (trough) on either side of the origin.

Domain and range The denominator $1+x^2$ is at least $1$ and never zero, so the function is defined for all real $x$. Because the extrema below are the global maximum and minimum, the range is $\left[-\dfrac{1}{2}, \dfrac{1}{2}\right]$.

Symmetry Since $f(-x) = \dfrac{-x}{1+x^2} = -f(x)$, the function is odd and its graph is point-symmetric about the origin, through which it passes at $(0,0)$.

Monotonicity and extrema By the quotient rule the derivative is $f'(x) = \dfrac{(1+x^2) - x\cdot 2x}{(1+x^2)^2} = \dfrac{1-x^2}{(1+x^2)^2}$. It is positive for $-1 < x < 1$ and negative elsewhere, so there is a maximum $\dfrac{1}{2}$ at $x = 1$ and a minimum $-\dfrac{1}{2}$ at $x = -1$ — the tops of the two bumps.

Asymptotes and limits As $x \to \pm\infty$ the $x^2$ in the denominator dominates and $y \to 0$, so the $x$-axis is a horizontal asymptote. The decay is a gentle $1/x$, so the curve returns slowly to zero beyond the bumps.

Inflection points The second derivative $f''(x) = \dfrac{2x(x^2 - 3)}{(1+x^2)^3}$ changes sign at $x = 0$ and $x = \pm\sqrt{3} \approx \pm 1.732$, giving three inflection points. The curve crosses most steeply at the origin, and $y = \dfrac{\sqrt{3}}{4} \approx 0.433$ at $x = \sqrt{3}$.

Relation to other functions It arises as the derivative of a logarithm: $\dfrac{d}{dx}\left[\dfrac{1}{2}\ln(1+x^2)\right] = \dfrac{x}{1+x^2}$.

Applications In physics, for the Lorentzian response describing resonance, the absorptive part is $\dfrac{1}{1+x^2}$ while the phase shift (dispersive part) takes exactly this form $\dfrac{x}{1+x^2}$. It serves as a model in signal processing and the frequency response of AC circuits, wherever a paired crest and trough appear.