Answer:
[tex](-\infty,-3)\cup(-3,\frac{4}{3})\cup(\frac{4}{3},\infty)[/tex]Explanation:
Given the function:
[tex]f(z)=\frac{3z+8}{3z^2+5z-12}[/tex]We want to find the domain of f(z).
The domain of a rational function is the set of values of z at which the denominator is not equal to 0.
To find the domain of f(z), set the denominator equal to 0 to find the excluded values.
[tex]\begin{gathered} 3z^2+5z-12=0 \\ 3z^2+9z-4z-12=0 \\ 3z(z+3)-4(z+3)=0 \\ (3z-4)(z+3)=0 \\ 3z-4=0,z+3=0 \\ z=-3,z=\frac{4}{3} \end{gathered}[/tex]The excluded values of the domain are -3 and 4/3.
Therefore, the domain of f(z) is:
[tex](-\infty,-3)\cup(-3,\frac{4}{3})\cup(\frac{4}{3},\infty)[/tex]
