Question

I need help with this question

Answer 1

-4x-2; all real numbers except x=6/5

Option B is correct.

EXPLANATION

The given functions are:

`f(x) = - 20 {x}^(2) + 14x + 12`

We factor to get,

`f(x) = - 2(10 {x}^(2) - 7x - 6)`

Split the middle term:

`f(x) = - 2(10 {x}^(2) + 5x - 12x - 6)`

`f(x) = - 2(5x(2{x} + 1)- 6(2x + 1))`

`f(x) = - 2(2{x} + 1)(5x - 6)`

and

`g(x) = 5x - 6`

`( (f)/(g) ) = (f(x))/(g(x))`

`( (f)/(g) ) = ( - 20x + 14x + 12)/(5x - 6)`

where 5x-6≠0

x≠6/5

`( (f)/(g) ) = \frac{- 2(2{x} + 1)(5x - 6)}{5x - 6}`

Cancel the common factors to get,

`( (f)/(g) ) = - 2(2{x} + 1)`

`( (f)/(g) ) = - 4{x} - 2`

Answer 2

Option B is correct

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.