Define a DT LTI system
$ y[n] = x[n+1] + x[n]\, $
Obtain the Unit Impulse Response h[n]
By definition, to obtain the unit impulse response from a system defined by $ y[n] = x[n]\, $, simply replace the $ x[n]\, $ by $ \delta[n]\, $.
$ h[n] = \delta[n+1] + \delta[n]\, $
Obtain the System Function $ F(z)\, $ of the System
$ F(z) = \sum^{\infty}_{m=-\infty} h[m]e^{jm\omega} \, $
$ F(z) = \sum^{\infty}_{m=-\infty} (\delta[m+1] + \delta[m])e^{jm\omega} \, $
$ F(z) = \sum^{\infty}_{m=-\infty} \delta[m+1]e^{jm\omega} + \delta[m]e^{jm\omega} \, $
Since the delta function is only valid when its input is zero,
$ F(z) = e^{-j\omega} + e^{0j\omega} \, $
$ F(z) = 1 + e^{-j\omega} \, $
