Answer
The value of $\left( f\circ g \right)\left( x \right)$ is $\frac{7x}{2-4x}$ and its domain is $\underline{\left( -\infty ,0 \right)\cup \left( 0,\frac{1}{2} \right)\cup \left( \frac{1}{2},\infty \right)}$
Work Step by Step
Calculate $\left( f\circ g \right)\left( x \right)$ as follows:
$\begin{align}
& \left( f\circ g \right)\left( x \right)=f\left( g\left( x \right) \right) \\
& =f\left( \frac{2}{x} \right) \\
& =\frac{7}{\frac{2}{x}-4} \\
& =\frac{7x}{2-4x}
\end{align}$
The domain is the set of values of $x$ for which the function $f(x)$ can be defined.
Since $\left( f\circ g \right)\left( x \right)$ cannot be defined when the denominator becomes zero:
$\begin{align}
& 2-4x=0 \\
& x=\frac{1}{2}
\end{align}$
So, $g\left( x \right)=\frac{2}{x}$ which means the function cannot be defined for $x=0$.
This implies that the function is defined for all the real values except $x=\frac{1}{2},0$.
Therefore, the domain of the function is $\left( -\infty ,0 \right)\cup \left( 0,\frac{1}{2} \right)\cup \left( \frac{1}{2},\infty \right)$.
