Study of (x²−1) over (x²+1)

Problem

Study the function $f(x)=\dfrac{x^2-1}{x^2+1}$ (all six steps, with graph).

Solution

Domain: $\mathbb{R}$ (denominator never zero). Symmetry: $f(-x)=f(x)$, an even function.

Sign and intersections: $f(0)=-1$; zeros for $x^2-1=0 \Rightarrow x=\pm 1$.

Asymptotes: $\lim_{x\to\pm\infty} f(x) = 1$: horizontal asymptote $y=1$ (no vertical asymptote).

First derivative: $f'(x)=\dfrac{2x(x^2+1)-(x^2-1)2x}{(x^2+1)^2}=\dfrac{4x}{(x^2+1)^2}$. It vanishes at $x=0$; $f'<0$ for $x<0$, $f'>0$ for $x>0$: minimum at $(0,-1)$. Range $[-1,\,1)$.

Second derivative: $f''(x)=\dfrac{4(1-3x^2)}{(x^2+1)^3}$, which vanishes at $x=\pm\frac{1}{\sqrt3}$: two inflection points. Concave upwards for $|x|<\frac{1}{\sqrt3}$.

Even curve, minimum at $(0,-1)$, horizontal asymptote $y=1$.

Links

Topics: Derivatives
Concepts: Asymptote · Maximum minimum · Rules of differentiation
Skills: Study a function · Sketch a graph
Exercise type: Study of a function