For complex numbers $z$, let \[f(z) = \left\{ \begin{array}{cl} z^{2}&\text{ if }z\text{ is not real}, \\ z+2 &\text{ if }z\text{ is real}. \end{array} \right.\]Find $f(i)+f(1)+f(-1)+f(-i)$.
+26367 question
\(\text{For complex numbers } \\ z, \text{let } \left[f(z) = \left\{ \begin{array}{cl} z^{2}&\text{ if }z\text{ is not real}, \\ z+2 &\text{ if }z\text{ is real}. \end{array} \right.\right] \\ \text{Find } f(i)+f(1)+f(-1)+f(-i).\)
\(\begin{array}{|l|clcrc|l|} \hline z = i && f(i) = z^2 = i^2 &=& -1 && \text{(z is not real)} \\ z = 1 && f(1) = z+2 = 1+2 &=& 3 && \text{(z is real)} \\ z = -1 && f(-1) = z+2 = -1+2 &=& 1 && \text{(z is real)} \\ z = -i && f(-i) = z^2 = (-i)^2 = i^2 &=& -1 && \text{(z is not real)} \\ \hline && f(i)+f(1)+f(-1)+f(-i) = -1+3+1-1 = 2 \\ \hline \end{array} \)
