Question

find the domain of the rational function

  -2(x^2-4)  divided by  3(x^2+4x+4).

Answer 1

The function is f(x ) = - 2(x ^2 - 4) / 3(x ^2 + 4x + 4).

Find the factors of numerator expression - 2(x ^2 - 4).

= - 2(x ^2 - 2^2)

Apply difference of squares : a² - b² = (a + b )(a - b ).

= - 2(x + 2)(x - 2)

Find the factors of denominator expression 3(x ^2 + 4x + 4).

= 3(x ^2 + 2x + 2x + 4)

= 3[x (x + 2) + 2(x + 2)]

= 3(x + 2)(x + 2).

f(x ) = - 2(x ^2 - 4) / 3(x ^2 + 4x + 4) = - 2(x + 2)(x - 2) / 3(x + 2)(x + 2) = - 2(x - 2) / 3(x + 2).

f(x) = - 2(x - 2) / 3(x + 2).

Domain excludes x - values that result in division by zero.

The function has an implied domain that consists of all real x other than x = - 2.This value is excluded from the domain because division by zero is undefined.