Function #1 | Mathematic | Olympiad

Symbol of pi

Hey friend! I am here and bring a problem to you.
This is the problem

The function $f(x) = \frac{x}{1 - 2^{x}} - \frac{x}{2}$ is

• a. an even but not odd function
• b. an odd but not even function
• c. a both even and odd function
• d. a neither even nor odd function

| Answer | Open only if you did it or you couldn't do it |

The Answer is A

| Explanation | Open only if you did it or you couldn't do it |

The solution is

The characteristic of even function is
$f(-x) = f(x)$

The characteristic of odd function is
$f(-x) = -f(x)$

It is easy to see that the domain of $f(x)$ is $(- \infty,0) \cup (0, + \infty)$. When $x \in (- \infty,0) \cup (0, + \infty)$ we have,

$\begin{eqnarray*} f(-x) &=& \frac{-x}{1 - 2^{-x}} - \frac{-x}{2} \\ \\ &=& \frac{-x . 2^{x}}{2^{x} - 1} + \frac{-x}{2} \\ \\ &=& \frac{x}{1 - 2^{x}} - x + \frac{x}{2} \\ \\ &=& \frac{x}{1 - 2^{x}} - \frac{x}{2} = f(x). \end{eqnarray*} $

Therefore, $f(x)$ is an even function, obviously not an odd function