The z - transform of a signal x[n]is given by 4z-3 + 3z-1

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The z - transform of a signal x[n]is given by
4z^-3 + 3z^-1 + 2 - 6z² + 2z³
It is applied to a system, with a transfer function H(z) = 3z_–1 – 2.
Let the output be y(n). Which of the following is true?

1. y(n) is non causal with finite support
2. y(n) is causal with infinite support
3. y(n) = 0; |n| > 3
4. Re[Y(z)]_z=e^jθ = - Re[Y(z)]_z=e^jθ;

Correct Option: A

Given: X(z) = 4 z^–3 + 3z^–1 + 2 – 6z² + 2z³
H(z) = 3z^–1 – 2,
output, y(n) = x(n) × h(n)
Taking z-tranform both sides,
Y(z) = H(z) X (z)
Y(z) = X(z) · H(z)
Y(z) = 12 z^–4 – 8z^–3 + 9z^–2 – 4 – 18z + 18z² – 4z³
As, y[n] ≠ 0 for n < 0
So we see that, we have z, z², z³ which depends on future input but finite terms. So y(n) is non casual with finite support.