y=x1+x2y = \dfrac{x}{1+x^2}

The Rational Function y=x1+x2y = \dfrac{x}{1+x^2}

y=x1+x2y = \dfrac{x}{1+x^2} is a rational function with numerator xx and denominator 1+x21+x^2. It can be viewed as the Witch of Agnesi multiplied by xx, and its defining feature is a single crest and a single trough on either side of the origin.

Domain and range

  • The domain is all real numbers
  • The range is −12≤y≤12-\dfrac{1}{2} \leq y \leq \dfrac{1}{2}
  • Odd function
  • Passes through the origin

The denominator is at least 11 and never zero, so the function is defined everywhere.

Symmetry

Since f(−x)=−f(x)f(-x) = -f(x), the function is odd and its graph is point-symmetric about the origin.

Monotonicity and extrema

By the quotient rule the derivative is the following.

f′(x)=(1+x2)−x⋅2x(1+x2)2=1−x2(1+x2)2f'(x) = \frac{(1+x^2) - x \cdot 2x}{(1+x^2)^2} = \frac{1-x^2}{(1+x^2)^2}
xx⋯\cdots−1-1⋯\cdots11⋯\cdots
f′f'−-00++00−-
ffdecreasingminimum −12-\dfrac{1}{2}increasingmaximum 12\dfrac{1}{2}decreasing

These are the tops of the two bumps, and since they are the global extrema they fix the range.

Asymptote

As x→±∞x \to \pm\infty the x2x^2 in the denominator dominates and y→0y \to 0, so the xx-axis is a horizontal asymptote. The decay is a gentle 1x\dfrac{1}{x}, so the curve returns slowly to zero beyond the bumps.

Inflection points

The second derivative f′′(x)=2x(x2−3)(1+x2)3f''(x) = \dfrac{2x(x^2 - 3)}{(1+x^2)^3} changes sign at three places.

xxyy
0000
3≈1.732\sqrt{3} \approx 1.73234≈0.433\dfrac{\sqrt{3}}{4} \approx 0.433
−3-\sqrt{3}−34-\dfrac{\sqrt{3}}{4}

The curve crosses most steeply at the origin.

Relation to other functions

It arises as the derivative of a logarithm.

ddx[12ln⁡(1+x2)]=x1+x2\frac{d}{dx}\left[\frac{1}{2}\ln(1+x^2)\right] = \frac{x}{1+x^2}

Applications

In physics, for the Lorentzian response describing resonance1, the absorptive part is 11+x2\dfrac{1}{1+x^2} while the phase shift, the dispersive part, takes exactly this form. It serves as a model in signal processing and in the frequency response of AC circuits, wherever a paired crest and trough appear.

  1. Resonance, Wikipedia