Answer
$\color{blue}{\bf{ \sqrt{5x-4} -\dfrac{1}{x} }}$
$\color{blue}{\bf{ \sqrt{5x-4} + \dfrac{1}{x} }}$
$\color{blue}{\bf{ \dfrac{ - \sqrt{ 5x-4 } }{x} }}$
$\color{blue}{\bf{ -x \sqrt{5x-4} }}$
$\color{blue}{\bf\text{domain: }{ [ \dfrac{4}{5} ,\infty) }}$
Work Step by Step
We are given the two functions $\bf{f}$ and $\bf{g}$
$\bf{f(x) = \sqrt{5x-4} }$ and $\bf{g(x) = -\dfrac{1}{x} }$
We are asked to find $\bf{( f+g )( x )}$ and its domain.
$f(x)+g(x)$
$\color{blue}{\bf{ \sqrt{5x-4} -\dfrac{1}{x} }}$
$x\neq0$ because $\dfrac{1}{0}$ is undefined
$5x-4\geq0$ or $ \sqrt{5x-4}$ would not be a real number
$5x\geq4$
$x\geq\dfrac{4}{5}$
Therefore the domain is: $\color{blue}{\bf{ [ \dfrac{4}{5} ,\infty) }}$
We are asked to find $\bf{( f-g )( x )}$ and its domain.
$f(x)-g(x)$
$\color{blue}{\bf{ \sqrt{5x-4} + \dfrac{1}{x} }}$
Its domain is the same as above, $\color{blue}{\bf{ [ \dfrac{4}{5} ,\infty) }}$
We are asked to find $\bf{( fg )( x )}$ and its domain.
$( \sqrt{5x-4} )( -\dfrac{1}{x} ) $
$\color{blue}{\bf{ \dfrac{ - \sqrt{ 5x-4 } }{x} }}$
Its domain is the same as above, $\color{blue}{\bf{ [ \dfrac{4}{5} ,\infty) }}$
We are asked to find $\bf{( \dfrac{f }{ g} )( x)}$ and its domain.
$\dfrac{ \sqrt{5x-4} }{ -\dfrac{1}{x} }$
$\dfrac{\sqrt{5x-4}}{1}\div-\dfrac{1}{x}$
$\dfrac{\sqrt{5x-4}}{1}\times-\dfrac{x}{1}$
$\color{blue}{\bf{ -x \sqrt{5x-4} }}$
Its domain is the same as above, $\color{blue}{\bf{ [ \dfrac{4}{5} ,\infty) }}$
