Question

Given f(x)=3x^2+10x−8 and g(x)=3x^2−2x .

What is (fg)(x)

Answer 1

`(x+4)/(x) \hspace{8}where\hspace{8}x\\eq0,(2)/(3)`

Explanation:

The division between functions is defined as:

`((f)/(g))(x)=(f(x))/(g(x)) ,\hspace{10}g(x)\\eq0`

So:

`((f)/(g) )(x)=(3x^2+10x-8)/(3x^2-2x)\\ \\Factor\hspace{3}x\hspace{3}out\hspace{3}the\hspace{3}denominator\\\\((f)/(g) )(x)=(3x^2+10x-8)/(x(3x-2))\\\\Factor\hspace{3}the\hspace{3}numerator\\\\((f)/(g) )(x)=(4(3x-2)+x(3x-2))/(x(3x-2))\\\\Factor\hspace{3}3x-2\hspace{3}from\hspace{3}the\hspace{3}numerator\\\\((f)/(g) )(x)=((3x-2)(x+4))/(x(3x-2))=(x+4)/(x)`

Since
`g(x) \\eq0` , let's find its roots:

`3x^2-2x=0\\\\Factor\\\\x(3x-2)=0\\\\Split\hspace{3}into\hspace{3}two\hspace{3}equations:\\\\(1):x=0\\(2):3x-2=0`

For (2)

`3x=2\rightarrow x=(2)/(3)`

Therefore the roots are:

`x=0,\hspace{3}x=(2)/(3)`

Finally the complete answer is:

`((f)/(g))(x)= (x+4)/(x) \hspace{8}where\hspace{8}x\\eq0,(2)/(3)`

Answer 2

the answer for this is C