RSA - source and image - all are possible?
If you're asking whether the power map $x\mapsto x^a \pmod n$ is injective, it suffices to take $a$ with $1 &lt; a &lt; \phi(n)$ and $\gcd(a,\phi(n)) = 1$.  

Edit:  I couldn't complete the answer earlier. As mentioned in the comments $gcd(a,n)$ is enough, but for computational efficiency one might as well take $a \pmod n$ as the exponent. In the RSA setup, $n$ is square free thus the function is bijective.

To see this, note that after all $e,d$ have similar roles and if one of them, say $e$, satisfies $\gcd(e,n)=1,$ then there is an integer $f$ such that $ed+fn=1$ (by the fact that they are inverses) thus demonstrating $gcd(d,n)=1.$)