Is it true that $"\exists f: A\to B, \ f\text{ bijective}" \iff "\exists f: B\to A, \ f\text{ bijective}"$ Is it true that 
$$"\exists f: A\to B, \quad f\quad bijective" \iff "\exists f: B\to A, \quad f\quad bijective"$$
Is it the same $f$? 
Intuitively I would say yes to both of these questions, but this example got me doubting: 
Let $f(x)=e^x.$ It is true that $"f:\mathbb{R}\to\mathbb{R}^+, \quad f\quad bijective"$, since every element in $\mathbb{R^+}$ can be reached (surjective) and from simply looking at the graph follows that it is injective. 
However, $"f:\mathbb{R^+}\to \mathbb{R}, \quad f \quad bijective"$ does not seem true since we can't reach the negative elements of $\mathbb{R}$ (not surjective).
What's going on? 

You need to take the inverse: if $f:A\to B$ is a bijection, then really it is the function $f^{-1}$ that goes backwards from $B$ to $A$. In your example, $f$ is the exponential function, and the inverse is $\log: \mathbb{R}^+ \to \mathbb{R}$. 
Hopefully this convinces you that your first question is yes: there is a bijection $f:A\to B$ if and only if there is one from $B\to A$; it just won't be the same function.
