+1859 For complex numbers $z$, the function $f(z)$ is defined by
\[f(z) = \left\{
\begin{array}{cl}
z + 1 &\text{ if }z\text{ is not real}, \\
z - i &\text{ if }z\text{ is real}.
\end{array}
\right.\]
Compute $f(i) + f(f(1)) + f(f(f(-1))) + f(f(f(f(-i))))$.
+177 Read the definition of the function and apply it to each input.
\(f(i) = i + 1 \\ f(f(1)) = f(1 - i) = 2 - i \\ f(f(f(-1))) = f(f(-1 - i)) = f(-i) = -i + 1 \\ f(f(f(f(-i)))) = f(f(f(-i + 1))) = f(f(-i + 2)) = f(-i + 3) = -i + 4\)
Now, find its sum.
\(\begin{align*} f(i) + f(f(1)) + f(f(f(-1))) + f(f(f(f(-i)))) &= i + 1 + 2 - i - i + 1 -i + 4 \\ &= -2i + 8 \end{align*}\)
