Monotonicity of x/(x²+1)

Problem

Say whether $f(x) = \dfrac{x}{x^2+1}$ is monotonic and justify.

Solution

No, it is not monotonic on all of $\mathbb{R}$. It is enough to exhibit an increasing piece and a decreasing one by comparing a few values: $f(0)=0, \qquad f(1)=\frac{1}{2}, \qquad f(2)=\frac{2}{5}=0{,}4, \qquad f(3)=\frac{3}{10}=0{,}3.$ Going from $x=0$ to $x=1$ the function increases ($0 \to \tfrac12$), but going from $x=1$ to $x=2$ the function decreases ($\tfrac12 \to 0{,}4$). Since on one piece it increases and on another it decreases, it is neither increasing nor decreasing on the whole domain: it is not monotonic. (It can be shown that it is increasing on $[-1,1]$ and decreasing elsewhere, with maximum $f(1)=\tfrac12$ and minimum $f(-1)=-\tfrac12$.)

Links

Topics: Functions and properties
Concepts: Monotonicity
Skills: Proving · Studying a function
Exercise types: Study of a function