Suppose $f : [0, 1] \to (0, 1)$ is a bijection. Using $f$, construct a bijection from $[−1, 1]$ to $\mathbb{R}$.
I don't know how to approach this question. I know there are similar questions, but none of those considered a case when a bijection is from a closed interval to $\mathbb{R}$. Please help me out. Thank you!
It's a simple task I'll leave to you to find a bijection $g: [-1,1]\to[0,1]$.
Now if we find a bijection $h: (0,1)\to\mathbb{R}$, then the map $h \circ f\circ g$ is what you are looking for.
For $h$, you should think about, for example, the tangens, which is a bijective map $\tan: (-\pi/2,\pi/2) \to \mathbb{R}$. By shifting and rescaling it in the right way, you will get the correct $h$.