find the domain in interval notation for the function

Question

f(x) 2x^3-250/x^2-2x-15

Answer 1

I will assume that you intended this to be a rational algebraic function, but you CARELESSLY OMITTED PARENTHESES REQUIRED surrounding each of numerator & denominator.

 

By factoring, I get f(x) = 2(x-5)(x²+5x+25) ÷ (x-5)(x+3) so I note that 5 and -3 are excluded from the domain.

there's a hole where x=5; x = -3 is a vertical asymptote. Now the identical factors (x-5) may be cancelled.

 

To the right of x=-3, the function is always positive, over the domain; to the left of -3, the function is always negative.

(observe that the numerator of the simplified function is always positive)

 

The domain is all reals except -3 and 5; V.A. at -3, HOLE at 5.

Answer 2

(-∝, 0) ∪ (0, ∝)