Problem in finding whether function is one-one or many-one

Question

Let $f : \mathbb{R}\rightarrow \mathbb{R} $, where $f(x) = 2^{|x|} - 2^{-x}$

How do we find whether is function is one to one or many to one?

Simplifying it we get $f(x) = \frac{2^{{|x|}+x} - 1}{2^x}$

It seems to be one-one, but the answer given is many-one. How to go about with this question?

Answer 1

Hint:

Consider what happens when $x<0$.

Also, we don't have to prove it generally, you can just evaluate at a few negative values.

Answer 2

HINT: When $f$ is equal to zero?

Answer 3

Take x=-1 and x=-2 and plug them into f.