Answer
$\left (\frac{f}{g} \right )(x)=\sqrt{x-5}$.
$[5,\infty)$.
Work Step by Step
The given functions are
$f(x)=\sqrt {x^2-25}$ and $g(x)=\sqrt {x+5}$
$\Rightarrow \left (\frac{f}{g} \right )(x)=\frac{f(x)}{g(x)}$
Substitute both functions.
$\Rightarrow \left (\frac{f}{g} \right )(x)=\frac{\sqrt {x^2-25}}{\sqrt {x+5}}$
Divide the radicands and retain the common index.
$\Rightarrow \left (\frac{f}{g} \right )(x)=\sqrt{\frac{ {x^2-25}}{x+5}}$
Factor $x^2-25$
$\Rightarrow x^2-5^2$
Use the special formula $A^2-B^2=(A+B)(A-B)$.
$\Rightarrow (x+5)(x-5)$
Back substitute the factor into the fraction.
$\Rightarrow \left (\frac{f}{g} \right )(x)=\sqrt{\frac{ {(x+5)(x-5)}}{x+5}}$
Cancel common terms.
$\Rightarrow \left (\frac{f}{g} \right )(x)=\sqrt{x-5}$
The domain is all positive real numbers greater than or equal to $5$.
The interval notation is $[5,\infty)$.
