Determing inverse Z-transform using impulse response?
When you have an $H(z)$ (a transfer function), its inverse Z-transform is the impulse response $h[n]$. iztrans is a symbolic function to compute any (causal) inverse Z-transform, either it is related to an impulse response or not. impz is a non-symbolic function to compute some samples of the impulse response of a digital filter.

If I were you and I wanted to find the inverse Z-transform of the given $H(z)$ - which is the impulse response -  I would use residuez:

[r,p,k] = residuez([4 3 9], [4 3 -4])
r =

 1.0544
 2.1956


p =

-1.4430
 0.6930


k =

 -2.2500


That allows you to write the impulse response as $$h[n] \approx -2.25\delta[n] + 1.0544(-1.443)^nu[n] + 2.1956(0.693)^nu[n] $$

given that we're talking about a causal system.

Alternatively, you can use

[h,n] = impz([4 3 9], [4 3 -4]);


but that will give you only a couple of samples of the impulse response.