Open intervals are connected using a continuous function

Question

I have this problem which I can't solve it using the suggestion given.

Using the continuous function $f(x)=$ $\frac{x}{1 + |x|}$ show that $(-1,1)$ is connected. Conclude that every open interval $(a,b)$ is connected.

I have solved this problem by a different way that what is suggested. I would like some hints to help me solve it using the given suggestion.What I can take for granted is that the rational numbers are not connected and that a closed interval is connected and the usual definitions of continuous functions.

If someone has already asked this or a similar answer please redirect me because I couldn't find it. Thanks in advance!

Answer 1

You can use two facts. The first one is that $\mathbb{R}$ is connected and the second one is that a continuous map preserves the connectedness.