# Domain, zeroes, and graphing to check the function

#### thebestrose828

Hey,
I really need help solving this question. If someone could provide a step-by-step guide? The question asks for the domain of the function, zeroes, and to graph it to check the function.
The function is as follows:
x3 +x2-2x
_________=f(x)
2x3-x2-6x

(The above is supposed to be those two equations divided by each other equals f(x), not really that computer savvy with typing math stuff)

Last edited:

#### skeeter

MHF Helper
start by factoring numerator & denominator ...

$f(x) = \dfrac{x(x+2)(x-1)}{x(2x+3)(x-2)}$

note that the numerator & denominator have a common factor of $x$ ... this indicates a point discontinuity (a "hole") at $(0,1/3)$, also indicating the absence of a y-intercept.

vertical asymptotes where $2x+3=0 \implies x = -\dfrac{3}{2}$ and where $x-2=0 \implies x = 2$

so, domain is all real values of $x$ except $x=0$, $x=-\dfrac{3}{2}$, and $x=2$

horizontal asymptote is determined by the ratio of the leading coefficients, $y = \dfrac{1}{2}$

zeros where $x+2=0 \implies x=-2$ and $x-1=0 \implies x=1$

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