Answer
$ 11-10i$
Work Step by Step
Given
\begin{equation}
f(x)=12 x-i \text { and } g(x)=6 x+3 i
\end{equation}
We want to find
\begin{equation}
(f g)\left(-\frac{1}{3}\right)
\end{equation}
We first compute the following
\begin{equation}
\begin{aligned}
f\left(-\frac{1}{3} \right)&=12\cdot \left( -\frac{1}{3} \right)-i\\
&=-4-i\\
&= -(4+i)\\
g\left(-\frac{1}{3} \right)&=6\cdot \left( -\frac{1}{3} \right)+3i\\
&= -2+3i\\
&= -(2-3i)
\end{aligned}
\end{equation}
It follows that:
\begin{equation}
\begin{aligned}
(fg)\left(-\frac{1}{3} \right)&=f\left(-\frac{1}{3} \right)\cdot g\left(-\frac{1}{3} \right) \\
&=[-(4+i)]\cdot [-(2-3i)] \\
&= (4+i)\cdot (2-3i) \\
& = 2(4+i)-3i(4+i)\\
& = 8+2i-12i-3i^2\\
& = 8-10i-3(-1)\\
& = 8-10i+3\\
&= 11-10i
\end{aligned}
\end{equation}
