Question

Determine the domain of the function f(x)=-x-10/-3x^3+5x^2+10x

Answer 1

$$f(x)=\dfrac{-x-10}{-3x^3+5x^2+10x}$$

$$-3x^3+5x^2+10x\not=0$$

$$-x(3x^2-5x-10)\not=0$$

Let $3x^2-5x-10=0$

$$D=(-5)^2-4(3)(-10)=145$$

$$x=\dfrac{5\pm\sqrt{145}}{2(3)}$$

$$x_1=\dfrac{5-\sqrt{145}}{6}, x_2=\dfrac{5+\sqrt{145}}{6}$$

Domain:

$$(-\infin, \dfrac{5-\sqrt{145}}{6})\cup(\dfrac{5-\sqrt{145}}{6}, 0)$$

$$\cup (0, \dfrac{5+\sqrt{145}}{6})\cup(\dfrac{5+\sqrt{145}}{6}, \infin)$$