Question

PLEASE HELP ASAP!!!

let f(x) = 2x^2 + x - 3 and g(x) = x - 1

find (g/f)(x) and state its domain.

Answer 1

`( (g)/(f) )(x) = (1)/( 2x+3)`

The domain is:

`x \\e1 \: or \: x \\e - (3)/(2)`

Step-by-step explanation

The given functions are:

`f(x) = 2 {x}^(2) + x - 3`

and

`g(x) = x - 1`

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

`( (g)/(f) )(x) = \frac{x - 1}{2 {x}^(2) + x - 3 }`

Factor the quadratic trinomial in the denominator.

`( (g)/(f) )(x) = \frac{x - 1}{2 {x}^(2) +3 x - 2x- 3 }`

`( (g)/(f) )(x) = (x - 1)/( x(2x+3) -1 (2x + 3 ))`

`( (g)/(f) )(x) = (x - 1)/( (2x+3) (x-1))`

Cancel the common factors,

`( (g)/(f) )(x) = (1)/( 2x+3) \: where \: x \\e1 \: or \: x \\e- (3)/(2)`