Let us fix a positive natural number $N$. When $i$ be is a natural number smaller than $N$, coprime with $N$, we let $\mu(i)$ be the unique number in ${1, \ldots, N-1}$ that is the multiplicative inverse of $i$ N-i$ modulo $N$. I would like to know what is the maximum, when $i$ is in the range of the numbers from $1$ to $N-1$ that are coprime with $N$, of the function $f(i):=i+\mu(i)$.

Let us fix a positive natural number $N$. When $i$ be a natural number smaller than $N$, coprime with $N$, we let $\mu(i)$ be the unique number in ${1, \ldots, N-1}$ that is the multiplicative inverse of $i$ modulo $N$. I would like to know what is the maximum, when $i$ is in the range of the numbers from $1$ to $N-1$ that are coprime with $N$, of the function $f(i):=i+\mu(i)$.