Question

Determine the domain of (f/g)(x) when f (x)=1/x and g (x)=sqrtx+5

Answer 1

we are given

$f(x)=\frac{1}{x}$

$g(x)=\sqrt{x+5}$

Calculation of (f/g)(x):

$(\frac{f}{g})(x)=\frac{f(x)}{g(x)}$

now, we can plug values

$(\frac{f}{g})(x)=\frac{\frac{1}{x}}{\sqrt{x+5}}$

now, we can simplify it

$(\frac{f}{g})(x)=\frac{1}{x\sqrt{x+5}}$

Domain:

we know that domain is all possible values of x for which any function is defined

function will be undefined when denominator =0

$x\sqrt{x+5} =0$

$x=-5,x=0$

we are given sqrt function

and we know that value inside sqrt will always be greater than 0

$x+5>0$

$x>-5$

so, domain will be

$(-5,0)U(0,\infty)$..............Answer