Answer:
Option B is correct
Step-by-step explanation:
Given:
f(x) = -20x^2 +14x +12 and
g(x) = 5x - 6
We need to find f/g and state its domain.
f/g = -20x^2 +14x +12/5x - 6
Taking -2 common from numerator:
f/g = -2(10x^2 - 7x - 6) / 5x -6
Factorize 10x^2 - 7x - 6= 10x^2 - 12x +5x -6
Putting in the above equation
f/g = -2(10x^2 - 12x +5x -6)/ 5x -6
f/g = -2(2x(5x-6) + 1 (5x-6)) / 5x-6
f/g = -2 ( (2x+1)(5x-6))/5x-6
cancelling 5x-6 from numerator and denominator
f/g = -2(2x+1)
f/g = -4x -2
The domain of the function is set of all values for which the function is defined and real.
So, our function g(x) = 5x -6 and domain will be all real numbers except x = 6/5 as denominator will be zero if x=5/6 and the function will be undefined.
So, Option B is correct.
ANSWER
-4x-2; all real numbers except x=6/5
Option B is correct.
EXPLANATION
The given functions are:
[tex]f(x) = - 20 {x}^{2} + 14x + 12[/tex]
We factor to get,
[tex]f(x) = - 2(10 {x}^{2} - 7x - 6)[/tex]
Split the middle term:
[tex]f(x) = - 2(10 {x}^{2} + 5x - 12x - 6)[/tex]
[tex]f(x) = - 2(5x(2{x} + 1)- 6(2x + 1))[/tex]
[tex]f(x) = - 2(2{x} + 1)(5x - 6)[/tex]
and
[tex]g(x) = 5x - 6[/tex]
[tex]( \frac{f}{g} ) = \frac{f(x)}{g(x)}[/tex]
[tex]( \frac{f}{g} ) = \frac{ - 20x + 14x + 12}{5x - 6}[/tex]
where 5x-6≠0
x≠6/5
[tex]( \frac{f}{g} ) = \frac{- 2(2{x} + 1)(5x - 6)}{5x - 6}[/tex]
Cancel the common factors to get,
[tex]( \frac{f}{g} ) = - 2(2{x} + 1)[/tex]
[tex]( \frac{f}{g} ) = - 4{x} - 2[/tex]
