Finding the frequency response of a filter defined as a Z-domain transfer function I am software engineer studying DSP on my own. I came across this question in a graduate textbook and am not sure how to proceed.

Determine the frequency response for the filter:
H(z) = 1.5 / (1 + 3z^-1)


Clearly, this is a z-domain transfer function. How do I get the frequency response? I am thinking of converting it into a time-domain difference equation, calculating about 128 points, running it through MATLAB's FFT, plot it, and voila, the final curve would be the answer. Is that a right approach? Is there a more analytic/math-y answer?
The analytic way is to substitute the variable $z$ by $e^{j\omega}$ to get the frequency response $H(\omega)$ (with $\omega = \frac{2 \pi f}{F_s}$) - that is to say, the frequency response is the $z$ transform evaluated on the unit circle.

Note that matlab has a built-in function for plotting the frequency response straight from filter coefficients (freqz), whose source code can be seen. Have a look! The implementation does not even involve solving the difference equation or filtering an impulse - you can simply divide the FFT of the numerator sequence by the FFT of the denominator sequence - padded up to the desired number of frequency points.