91Ó°ÊÓ

given,

A graph is given in the question with the complete graph details, we have to find a rational function of the given graph.

The numerator of a rational function $R\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)}$in the lowest terms determines the $x-$intercepts of its graph. The graph shown in Figure 47 has $x-$intercepts 1 (odd multiplicity; graph crosses the $x-$axis) and 3 (odd multiplicity; graph crosses the $x-$axis). So one possibility for the numerator is$p\left(x\right)=$$(x-1)(x-3)$

The denominator of a rational function in the lowest terms determines the vertical asymptotes of its graph. The vertical asymptotes of the graph are $x=-1andx=2$, Since width="33" height="20" role="math">R(x) approaches $∞$ from both sides of $x=-1$

So, $(x+1)$is a factor of even multiplicity in $q\left(x\right)$.

Also, $R\left(x\right)$approaches $-∞$from both sides of $x=2$.

So, $(x-2)$is a factor of even multiplicity in $q\left(x\right)$. A possibility for the denominator is $q\left(x\right)=$${(x+1)}^2{(x-2)}^2$

So far we have,