Question

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

Answer 1

The domain is: {x > -5 and x≠0 }

Explanation:

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

`f(x)=(1)/(x)`

and

`g(x)=√(x+5)`

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

`((f)/(g))(x)=(f(x))/(g(x))`

This means that:

`((f)/(g))(x)=((1)/(x))/(√(x+5))`

i.e.

`((f)/(g))(x)=(1)/(x√(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