Question 1204539
Given the Rational Function f(x)= p(x)/q(x), which of the following statements are true? Check all that apply.

Asymptotes
A vertical asymptote of a graph is a vertical line x = a where the graph tends toward positive or negative infinity as the inputs approach a.


<font color=red>recall</font>:

If{{{ N}}} is the degree of the numerator and {{{D }}}is the degree of the denominator, and

if {{{N < D}}}, then the horizontal asymptote is {{{y = 0}}}
if {{{N = D}}}, then the horizontal asymptote is {{{y}}} = ratio of the leading coefficients.
if {{{N > D}}}, then there is {{{no}}} horizontal asymptote.
if {{{N > D+1}}} ( the degree of numerator is {{{1}}} more than the degree of the denominator) ,then there is slant asymptote 

A horizontal asymptote of a graph is a horizontal line {{{y = b}}} where the graph approaches the line as the inputs approach {{{infinity}}} or {{{-infinity}}} .

A slant asymptote of a graph is a slanted line {{{y = mx + b }}}where the graph approaches the line as the inputs approach {{{infinity}}} or {{{-infinity}}}.


a. If q(x) has a higher degree term than p(x) then the Horizontal Asymptote is y=0

<font color=red>True</font>

reason:
if {{{N < D}}}, then the horizontal asymptote is {{{y = 0}}}


b. If the highest degree term in p(x) is greater than the highest degree term in q(x) there will be more than one Horizontal Asymptote

<font color=red>False</font>

reason:
if{{{ N > D}}}, then there is {{{no }}}horizontal asymptote.



c. If the highest degree term of p(x) is the same as the highest degree term of q(x) then the Horizontal Asymptote is x=0

<font color=red>False</font>

reason:
if {{{ N =D}}}, then the horizontal asymptote is {{{y}}} = ratio of leading coefficients

d. If f(x) has a HORIZONTAL ASYMPTOTE y=a, then as the input values increase or decrease without bound, the output values will approach a.

<font color=red>True</font>

example

{{{f(x)=(6x-1)/(3x+7)}}} horizontal asymptote of {{{(6 x - 1)/(3 x + 7)->2}}} as {{{x}}}-> ± {{{infinity}}}


e. The Horizontal Asymptote is a guiding line for the function as the input values increase or decrease without bound

<font color=red>True</font>