Question

What is the domain of `f(x)=x/(x^3+8)`?

Answer 1

Domain: `(-oo, -2) uu (-2, + oo)`

Explanation:

You need to exclude from the function's domain any value of `x` that would make the denominator equal to zero.

This means that you need to exclude any value of `x` for which

`x^3 + 8 = 0`

This is equivalent to

`x^3 + 2""^3 = 0`

You can factor this expression by using the formula

`color(blue)(a^3 + b^3 = (a+b) * (a^2 - ab + b^2))`

to get

`(x+2)(x^2 - 2x + 2^2) = 0`

`(x+2)(x^2 - 2x + 4) = 0`

This equation will have three solutions, but only one will be real.

`x+2 = 0 implies x_1 = -2`

and

`x^2 - 2x + 4 = 0`

`x_(2,3) = (-(2) +- sqrt((-2)^2 - 4 * 1 * 4))/(2 * 1)`

`color(red)(cancel(color(black)(x_(2,3) = (2 +- sqrt(-12))/2))) ->` produces two complex roots

Since these two roots will be complex numbers, the only value of `x` that must be excluded from the function's domain is `x=-2`, which means that, in interval notation, the domain of the function will be `(-oo, -2) uu (-2, + oo)`.