Question

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

Answer 1

Answer:

all work is answer and shown and pictured

Answer 2

Answer:

The domain is:   {x > -5 and x≠0 }

Step-by-step explanation:

We are given a function f(x) and g(x) as follows:

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

and

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

Now, the function (f/g)(x) is given by:

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

This means that:

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

i.e.

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

Now we know that a square root function is defined if the radicand is positive.

i.e.

Here,

x+5 ≥ 0

i.e.

x ≥ -5

Also,

x≠0 and √(x+5)≠0

( otherwise the denominator will be zero and hence the expression will be not defined)

Hence, we have:

x≠0 and x≠ -5

This means that:

 x > -5 and x≠0