Answer: [tex](f/g)(x)=\frac{2x}{x+5}[/tex]
Step-by-step explanation:
Given the function f(x):
[tex]f(x)=4x^2+6x[/tex]
And the function g(x):
[tex]g(x)=2x^2+13x+15[/tex]
To find [tex](f/g)(x)[/tex] you need to divide the function f(x) by the function g(x).
Therefore, knowing this, you get:
[tex](f/g)(x)=\frac{4x^2+6x}{2x^2+13x+15}[/tex]
You can simplify the numerator by factoring out 2x:
[tex](f/g)(x)=\frac{2x(2x+3)}{2x^2+13x+15}[/tex]
You have to simplify the denominator:
Rewrite the term 13x as a sum of two terms whose product be 30:
[tex](f/g)(x)=\frac{2x(2x+3)}{2x^2+(10+ 3)x+15}[/tex]
Apply Distributive property:
[tex](f/g)(x)=\frac{2x(2x+3)}{2x^2+10x+ 3x+15}[/tex]
Make two groups of two terms:
[tex](f/g)(x)=\frac{2x(2x+3)}{(2x^2+10x)+ (3x+15)}[/tex]
Factor out 2x from the first group and 3 from the second group:
[tex](f/g)(x)=\frac{2x(2x+3)}{(2x(x+5))+ 3(x+5)}[/tex]
Factor out (x+5):
[tex](f/g)(x)=\frac{2x(2x+3)}{(2x+3)(x+5)}[/tex]
Simplifying, you get:
[tex](f/g)(x)=\frac{2x}{x+5}[/tex]
ANSWER
[tex]( \frac{f}{g} )(x) = \frac{2x }{x + 5}[/tex]
where
[tex]x \ne - \frac{3}{2} \: or \: x = - 5[/tex]
EXPLANATION
The given functions are:
[tex]f(x) = 4 {x}^{2} + 6x[/tex]
and
[tex]g(x) =2 {x}^{2} + 13x + 15[/tex]
We want to find ,
[tex]( \frac{f}{g} )(x) = \frac{f(x)}{g(x)} [/tex]
[tex]( \frac{f}{g} )(x) = \frac{4 {x}^{2} + 6x }{2 {x}^{2} + 13x + 15} [/tex]
[tex]( \frac{f}{g} )(x) = \frac{2x(2x + 3) }{2{x}^{2} + 10x +3x + 15} [/tex]
[tex]( \frac{f}{g} )(x) = \frac{2x(2x + 3) }{2{x}(x + 5) +3(x + 5)} [/tex]
[tex]( \frac{f}{g} )(x) = \frac{2x(2x + 3) }{(2x + 3)(x + 5)} [/tex]
We cancel out the common factors to get:
[tex]( \frac{f}{g} )(x) = \frac{2x }{x + 5} [/tex]
where
[tex]x \ne - \frac{3}{2} \: or \: x = - 5[/tex]
