How to take z-transform of a function with an absolute exponent value?
You can simplify the expression by segmenting it to two intervals 


\$ u(n)\$  \$\space\space\space\space\space\$  for \$ n&gt;0 \$ 
\$ u(-n-1)\$ \$\space\space\$  for \$n &lt; 0\$


$$h(n) = (\frac{1}{2})^{n-1}u(n) + (\frac{1}{2})^{n}u(n) +(\frac{1}{2})^{-n+1}u(-n-1) + (\frac{1}{2})^{-n}u(-n-1) $$

According to the superposition theorem, z-transform of two or more functions is the same as taking the z-transform of each one individually, and then adding them. Same Applies here.



Lastly, you need to determine region of convergence