Question

Suppose that the functions f and g are defined as follows F(x)=5/x+7g(x)=2/xFind f/g then give its domain using an interval or union of intervals Simplify your answer as much as possible (f/g)(x)=Domain of f/g:

Answer 1

Given the functions:

`\begin{gathered} f(x)=(5)/(x+7) \\ \text{AND} \\ g(x)=(2)/(x) \end{gathered}`

Let's solve for the following:

• (a) f/g

To solve for f/g let's divide f(x) by g(x).

We have:

`(f)/(g)=(f(x))/(g(x))=((f)/(g))(x)=((5)/(x+7))/((2)/(x))`

Solving further, we have:

`\begin{gathered} ((f)/(g))(x)=(5)/(x+7)\ast(x)/(2) \\ \\ ((f)/(g))(x)=(5x)/(2(x+7)) \end{gathered}`

Therefore, the function f/g is:

`((f)/(g))(x)=(5x)/(2(x+7))`

• (b) Domain of f/g.

The domain is the set of all possible x-values where the function is defined.

To find the domain, set the denominator to zero and solve for x.

We have:

`2(x+7)=0`

Divide both sides by 2:

`\begin{gathered} (2(x+7))/(2)=(0)/(2) \\ \\ (x+7)=0 \end{gathered}`

Subtract 7 from both sides:

`\begin{gathered} x+7-7=0-7 \\ \\ x=-7 \end{gathered}`

Therefore, the domian is:

`\mleft(-\infty,-7\mright)\cup(-7,\infty)`

ANSWER:

`(a)\text{ ( }(f)/(g))(x)=(5x)/(2(x+7))`
`(b)\text{ Domain: }(-\infty,-7)\cup(-7,\infty)`