Is $\ln\left( \frac{1+2x}{1-2x}\right)$ an odd or even function?

Question

Is $$f:\left(-\frac{1}{2},\frac{1}{2}\right) \to \mathbb{R}, \ x \mapsto \ln\left( \frac{1+2x}{1-2x}\right)$$ an odd or even function?

The function can be decomposed this way:

$$f(x)=\ln(1+2x)-\ln(1-2x)$$

It seems that this function is an odd function but I dont know how to justify it.

Answer 1

A odd function must satisfy the equation $f(-x) = -f(x)$.

Let $x \in \left(-\frac{1}{2}, \frac{1}{2}\right)$. Then we have, as you rightly observed \begin{align*} -f(x) = \ln(1 - 2x) - \ln(1 + 2x) = \ln(1 + 2(-x)) - \ln(1 - 2(-x)) = f(-x). \end{align*}

Note that a function being odd graphically corresponds to rotational symmetry. Similarly to the graph of $x \mapsto x^3$ (the standard example in my opinion), when you rotate the graph 180° with respect to the origin (alternatively flip over the $y$- and then the $x$-axis), you end up with the same graph.