G(x) = -2x^3 – 15x^2 + 36x - state the intervals that the function is positive/negative. - state if the function has even, odd, or no symmetry. - state the domain and range? (state your answers in inequality and interval notation)

Consider the function First, factor it:

The x-intercepts are at points

1. From the attached graph you can see that

2. Since

the function is neither even nor odd.

3. The domain is the range is

a) G(x) factors as x(-2x^2 -15x+36) so has a zero at x=0 and at values of x revealed by the quadratic formula:

For ax²+bx+c=0, the solutions are

... x = (-b±√(b²-4ac))/(2a)

Here, we have a=-2, b=-15, c=36, so

... x = (15±√(225+288))/(-4) = -3.75±√32.0625

... x ≈ {-9.412, 1.912}

The function is positive for x in (-∞, -9.412) ∪ (0, 1.912).

The function is negative for x in (-9.412, 0) ∪ (1.912, ∞).

b) The function contains both even-degree and odd-degree terms, so has no even or odd symmetry. (A cubic always has odd symmetry about its point of inflection, but that point is not x=0 for this function.)

c) The domain and range of any odd-degree polynomial are all real numbers.

what shape is