Question

Find the domain of (f/g)(x) when f(x)=1/x and g(x)=square root x+8

Answer 1

Answer:

The domain of the given function is:

              $x>-8\ \text{and}\ x\neq 0$

Step-by-step explanation:

Domain of a function--

The domain of a function is the set of all the x-values i.e. the value of the independent variable for which a function is well defined.

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

$f(x)=\dfrac{1}{x}\\\\g(x)=\sqrt{x+8}$

Hence,

$(\dfrac{f}{g})(x)=\dfrac{1}{x\sqrt{x+8}}$

Now, we know that this function is well defined when the denominator quantity is defined and also when the denominator quantity is non-zero.

This means that:

          $x\neq 0$

and

$\sqrt{x+8}\neq 0$

i.e.

$x\neq -8$

Now we know that the square root function is defined when the radicand is greater than zero.

i.e.

$x+8\geq 0\\\\x\geq -8$

But $x\neq -8$

This means that:

$x>-8$

Hence, the domain is:

                       $x>-8\ \text{and}\ x\neq 0$

Answer 2

Step-by-step explanation:

all work is pictured/shown