Question

In Problems 45-48, find a rational function that might have the given graph. (More than one answer might be possible.)

Short answer

The function is$R\left(x\right)=\frac{x^2}{x^2-4}$

Step-by-step solution

Given information

Given the graph

Checking the asymptote

Here there are two asymptotes namely

$$\begin{gathered} x=-2 \\ x=2 \end{gathered}$$so the denominator is of the form$R\left(x\right)=\frac{A\left(x\right)}{(x+2)(x-2)}$

Checking the curve

The curve touches the $x$axis at the origin so it have even multiplicity of two. Then the horizontal asymptote is one so the values of leading coefficient should be same i.e. one .So the function is

$R\left(x\right)=\frac{x^2}{x^2-4}$