Question

Math part 2 question 4

Answer 1

`(x)/(x + 1)\\\\\text{which is the first answer choice }`

Explanation:

We are given

`f(x) = x^2 - x\\g(x) = x^2 - 1\\\\\text{and we are asked to find $ \left((f)/(g)\right)\left(x\right)$}`

`\left((f)/(g)\right)\left(x\right) = (f(x))/(g(x))\\\\\\= (x^2-x)/(x^2 - 1)`

`x^2 - x = x(x - 1)\text{ by factoring out x}\\\\x&2 - 1 = (x + 1)(x - 1) \text{ using the relation $a^2 - b^2 = (a + 1)(a - 1)$}`

Therefore,

`(x^2-x)/(x^2 - 1) = (x(x-1))/((x + 1)(x - 1))`

x - 1 cancels out from numerator and denominator with the result

`(x)/(x+1)`

So

`\left((f)/(g)\right)\left(x\right)$} = (x)/(x + 1)`