Consider a causal LTI system described by the difference equation

$y[n]+\frac{1}{2}y[n-1]=x[n]$

$($a$)$ Determine the frequency response $H(e^j)$ of this system.

$($b$)$ What is the response of the system to $x[n]=[n]-\frac{1}{2}[n-1]$

$($c$)$ Find the response to the inputs with the Fourier transform $x(e^j)=\frac{1}{(1-\frac{1}{1}{e}^{-})(1+\frac{1}{2}{e}^{-j})}$