Question

The domain of the given function is ____ (use integers or fractions for any numbers in the expression.)f(z) = 3z + 8 —————- 3z² +5z-12

Answer 1

`(-\infty,-3)\cup(-3,(4)/(3))\cup((4)/(3),\infty)`

Explanation:

Given the function:

`f(z)=(3z+8)/(3z^2+5z-12)`

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.

`\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=(4)/(3) \end{gathered}`

The excluded values of the domain are -3 and 4/3.

Therefore, the domain of f(z) is:

`(-\infty,-3)\cup(-3,(4)/(3))\cup((4)/(3),\infty)`