Question 960716
f is undefined at x=0, because denominators must not become zero.


You could make x y table of values, but you could also condense the function expression and evaluate a few things about it.


First notice that as x goes unbounded to the left or the right, f(x) will approach 4x, so this will be a slant asymptote, of y=4x.


{{{4x-3/x}}}
{{{4x(x/x)-3/x}}}
{{{highlight((4x^2-3)/x)}}}


Where is the numerator 0?
{{{4x^2=3}}}
{{{x^2=3/4}}}
{{{x=3/2}}} or {{{x=-3/2}}}, so these are critical points, as is x=0.


The three critical x values form intervals:
{{{-infin<x<-3/2}}}
{{{-3/2<x<0}}}
{{{0<x<3/2}}}
{{{3/2<x<infin}}}
Evaluate the sign of f in each interval.  That will help to determine if f is above or below the horizontal axis.



{{{graph(300,300,-10,10,-10,10,4x-3/x)}}}


A vertical asymptote in addition to the slant asymptote, is x=0.