Answer if each of the following functions is a bijection onto its range. For any function that is a bijection, identify $f^{-1}(5)$. Justify all of your answers.
a) $f(n)$ = $2n$ mod 10. The domain is $\mathbb{Z}_{10} = \{0,1,2,3,4,5,6,7,8,9\}$ Ok so this is not a bijection thank you T.Bongers.
b) $f(n)$ = $2n$ mod 11. The domain is $\mathbb{Z}_{11} = \{0,1,2,3,4,5,6,7,8,9,10\}$
Just want to confirm is (b) a bijection?
I will suggest the proof for the first one. For domains as small as this, it's easy enough to just write out the images for each $n$ if you don't see how to solve it in general; so let's start with
\begin{align*} f(0) &= 0\\ f(1) &= 2\\ f(2) &= 4\\ f(3) &= 6\\ f(4) &= 8\\ f(5) &= 0\\ f(6) &= 2\\ &\vdots\\ f(9) &= 8\\ \end{align*}
So we see that the image set is $\{0, 2, 4, 6, 8\}$, or the set of even numbers less than $10$. This is exactly what you'd expect from a function that doubles numbers and looks at the last digit. In particular, we see that it's neither an injection nor a surjection.