Question 976058
Like this:
f(x) = (x^2-2x-3)/(x^2+2x+3)


Which will properly render as {{{f(x)=(x^2-2x-3)/(x^2+2x+3)}}}.


Factor as much as possible.


{{{((x-3)(x+1))/(x^2+2x+3)}}}


Only two critical x values.  Denominator is positive everywhere and has no real roots, so no
vertical asymptotes.  Numerator gives f(x) its roots, which are  3 and -1.  The numerator is
equivalent itself to a parabola, and in this example, f is negative between the roots and positive
outside the interval which the roots bind.


Domain:  All real numbers.


The function f(x) is not a parabola.  There is a minimum, and a horizontal asymptote.  (Degree is two
for both numerator and denominator).


{{{graph(,,,,,,Factor as much as possible.


{{{((x-3)(x+1))/(x^2+2x+3)}}}


Only two critical x values.  Denominator is positive everywhere and has no real roots, so no
vertical asymptotes.  Numerator gives f(x) its roots, which are  3 and -1.  The numerator is
equivalent itself to a parabola, and in this example, f is negative between the roots and positive
outside the interval which the roots bind.


Domain:  All real numbers.


The function f(x) is not a parabola.  There is a minimum, and a horizontal asymptote.  (Degree is two
for both numerator and denominator).


{{{graph(300,300,-5,5,-5,5,(x^2-2x-3)/(x^2+2x+3))}}}