91Ó°ÊÓ

Consider the given question,

The denominator of a rational function in lowest terms determines the vertical asymptotes of its graph.

The vertical asymptotes are $x=-3,4$.

As $R\left(x\right)$approaches $∞$to the left of $x=-3$and $R\left(x\right)$approaches $-∞$to the right of localid="1646072581381" $x=-3$, then $x+3$is a factor of odd multiplicity of $q\left(x\right)$.

Similarly, localid="1646073819637" $\left(x-4\right)$is also a factor of odd multiplicity in $q\left(x\right)$.

So, one possibility for the denominator is given below,

$q\left(x\right)={\left(x-4\right)}^2\left(x+3\right)$.

Consider the given question,

The horizontal asymptote of the given graph is $y=3$. Thus, the degree of the numerator must be equal to the degree of the denominator and that the quotient of the leading coefficients must be $3$.

Hence, a possible rational function is given below,

localid="1646145213137" $R\left(x\right)=\frac{3\left(x+2\right){\left(x-1\right)}^2}{\left(x+3\right){\left(x-4\right)}^2}$