The domain of the given function is:
[tex]x>-8\ \text{and}\ x\neq 0[/tex]
Domain of a function--
The domain of a function is the set of all the x-values i.e. the value of the independent variable for which a function is well defined.
We have a function f(x) and g(x) as follows:
[tex]f(x)=\dfrac{1}{x}\\\\g(x)=\sqrt{x+8}[/tex]
Hence,
[tex](\dfrac{f}{g})(x)=\dfrac{1}{x\sqrt{x+8}}[/tex]
Now, we know that this function is well defined when the denominator quantity is defined and also when the denominator quantity is non-zero.
This means that:
[tex]x\neq 0[/tex]
and
[tex]\sqrt{x+8}\neq 0[/tex]
i.e.
[tex]x\neq -8[/tex]
Now we know that the square root function is defined when the radicand is greater than zero.
i.e.
[tex]x+8\geq 0\\\\x\geq -8[/tex]
But [tex]x\neq -8[/tex]
This means that:
[tex]x>-8[/tex]
Hence, the domain is:
[tex]x>-8\ \text{and}\ x\neq 0[/tex]
