Z-transform of an FIR filter QUESTION 

Compute the Z-transform of $y[n] = x[n] + 2x[n-1]$. and find the poles and zeros.

I just bombed an interview where I couldn't do this (because I have no grounding in fundamentals and have worked all from cookbooks). I'm not too bummed because I would have been working at something I wasn't prepared for. You get one guess who would ask this kind of question in a first phone interview.

ANSWER (30 minutes post interview):

$x[n-1] = z^{-1}$

$1 + 2z^{-1}$.

Why? 

Because the z-transform of an FIR is the coefficients of the impulse response multiplied by the delays.
If you know (or better: understand) the basic $\mathcal{Z}$-transform relation

$$x[n-k]\Longleftrightarrow z^{-k}X(z)$$

then from the given time domain equation you immediately get

$$Y(z)=X(z)+2X(z)z^{-1}=X(z)(1+2z^{-1})$$

Since the transfer function $H(z)$ is the ratio of $Y(z)$ and $X(z)$, you get

$$H(z)=1+2z^{-1}$$