How to detect if a function has symmetries

Question

let's say we have the function $$y = \frac{x}{x^2+1}$$

we see that y': $$y' =\frac{-x^2+1}{(x^2+1)^2}$$

by the second derivative test, we see that the points $x=1$ is a local maximum and $x=-1$ is a local minimum that's the only clue I have about the symmetry of this graph.

Answer 1

We can check whether the function is even or odd:

• If function is odd, meaning $f(-x)=-f(x)$ than it is symmetrical with respect to the origin
• If function is even, meaning $f(-x)=f(x)$ than it is symmetrical with respect to $y$-axis

In this example, we can see that $$f(-x)=\frac{-x}{x^2+1}=-f(x)$$ thus this function is odd, meaning it is symmetrical with respect to the origin.

Answer 2

Notice that

\begin{align*} f(x) = \frac{x}{x^{2} + 1} \Rightarrow f(-x) = -\frac{x}{(-x)^{2} + 1} = -\frac{x}{x^{2} + 1} = -f(x) \end{align*}

hence we conclude its graph is symmetric about the origin.