Question

Suppose that the functions f and f are defined as follows F(x)= 5/x g(x)=9/x+1 Find g/f then give its domain using an interval or union of intervals Simplify your answers (g/f)(x)=Domain of g/f:

Answer 1

Given:

$\begin{gathered} f(x)=\frac{5}{x} \\ g(x)=\frac{9}{x+1} \end{gathered}$

To find

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

We know that,

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

So,

$\begin{gathered} (\frac{g}{f})(x)=\frac{\frac{9}{x+1}}{\frac{5}{x}} \\ (\frac{g}{f})(x)=\frac{9}{x+1}\times\frac{x}{5} \\ (\frac{g}{f})(x)=\frac{9x}{5(x+1)} \end{gathered}$

Hence, the answer is,

$(\frac{g}{f})(x)=\frac{9x}{5(x+1)}$

Domain of the above function is,

Let the dinominator equals to 0, we get

x+1=0

x= -1

Hence, the domain of the function is,

$(-\infty,-1)\cup(-1,\infty)$