Question

PLEASE HELP ASAP!!!

Answer 1

ANSWER

$( \frac{g}{f} )(x) = \frac{1}{ 2x+3}$

The domain is:

$x \ne1 \: or \: x \ne - \frac{3}{2}$

EXPLANATION

The given functions are:

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

and

$g(x) = x - 1$

$( \frac{g}{f} )(x) = \frac{g(x)}{f(x)}$

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

Factor the quadratic trinomial in the denominator.

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

$( \frac{g}{f} )(x) = \frac{x - 1}{ x(2x+3) -1 (2x + 3 )}$

$( \frac{g}{f} )(x) = \frac{x - 1}{ (2x+3) (x-1)}$

Cancel the common factors,

$( \frac{g}{f} )(x) = \frac{1}{ 2x+3} \: where \: x \ne1 \: or \: x \ne- \frac{3}{2}$