Determine filter given transfer function I am given $$H(z) = 1 + \frac{\alpha}{1-\alpha z^{-1}}$$ where alpha is between $0$ and $1$. This is apparently is a low-pass filter with cutoff frequency $f_c$. 

How can I see that? And how can I compute/plot the frequency response in Matlab? Neither freqz(1+alfa,[1 -alfa]) nor abs(H(z)) seems to yield a low-pass filter on the above transfer function in Matlab.

Furthermore, $1-H(z)$ should be a high-pass filter. But again, using freqz(-alfa,[1 -alfa]) in matlab it is clearly not a high-pass filter.
No $$H(z) = 1 + \frac{\alpha}{1-\alpha z^{-1}}$$ is not a lowpass filter. It's a lowpass filter in parallel with an all pass filter (kind of low-boost shelving filter). If you want a lowpass filter which rejects the high frequencies then you may take the following:

$$H_{lp}(z) = \frac{\alpha}{1-\alpha z^{-1}}$$ which is a lowpass filter for the range of parameter $\alpha$. And therefore $1 - H_{lp}(z)$ will be the complementary highpass filter in simplest terms.

So you can see it by the following code:

freqz(alfa, [1 -alfa])


Note that the gain of the filter will be dependent on $\alpha$! Note also that as Matt.L commented you might have $1 + \alpha$ in the numerator instead...