What is the domain of f(x)=x/(x^3+8) f(x)=xx3+8?
1 Answer
Domain:
Explanation:
You need to exclude from the function's domain any value of
This means that you need to exclude any value of
x^3 + 8 = 0x3+8=0
This is equivalent to
x^3 + 2""^3 = 0x3+23=0
You can factor this expression by using the formula
color(blue)(a^3 + b^3 = (a+b) * (a^2 - ab + b^2))a3+b3=(a+b)⋅(a2−ab+b2)
to get
(x+2)(x^2 - 2x + 2^2) = 0(x+2)(x2−2x+22)=0
(x+2)(x^2 - 2x + 4) = 0(x+2)(x2−2x+4)=0
This equation will have three solutions, but only one will be real.
x+2 = 0 implies x_1 = -2x+2=0⇒x1=−2
and
x^2 - 2x + 4 = 0x2−2x+4=0
x_(2,3) = (-(2) +- sqrt((-2)^2 - 4 * 1 * 4))/(2 * 1)x2,3=−(2)±√(−2)2−4⋅1⋅42⋅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
