Domain, zeroes, and graphing to check the function

Jan 2017
25
0
Canada
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)
Thank you for your help!!
 
Last edited:

skeeter

MHF Helper
Jun 2008
16,217
6,765
North Texas
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 $x-1=0 \implies x=1$
 

Attachments