find the domain of (x+2)/x^3-16x
all x ^ 3 different of zero
this mean :
all x different of zero
(- infinity , 0 ) U ( 0 , infinity )
assuming the usual carelessness with parentheses, I see this as
(x+2)/(x^3-16x)
(x+2)/(x(x-4)(x+4))
so the domain is all reals except 0, -4, 4
To find the domain of a rational function, we need to determine the values of x that make the denominator equal to zero. When the denominator is zero, the function is undefined.
In this case, the denominator is x^3 - 16x. To find the values of x that make the denominator zero, we set it equal to zero and solve for x:
x^3 - 16x = 0
Factoring out an x from both terms:
x(x^2 - 16) = 0
Now, we have two factors: x and x^2 - 16. For the product to be zero, at least one of the factors must be zero.
Setting x = 0:
x = 0
Now, setting x^2 - 16 = 0:
x^2 = 16
Taking the square root of both sides:
x = ±√16
Simplifying:
x = ±4
Therefore, the values of x that make the denominator zero are x = 0 and x = ±4.
The domain of the function is all real numbers except for x = 0 and x = ±4. So, the domain can be expressed as:
Domain: (-∞, -4) ∪ (-4, 0) ∪ (0, 4) ∪ (4, ∞)