which of the following could be the function graphed?\n$f(x)=\frac{8x^{2}}{x^{2}-9}$\n$f(x)=\frac{119x}{792x…

Explanation:

Step1: Analyze vertical - asymptotes

For a rational function $f(x)=\frac{N(x)}{D(x)}$, vertical asymptotes occur at the values of $x$ that make $D(x) = 0$. For $f(x)=\frac{8x^{2}}{x^{2}-9}=\frac{8x^{2}}{(x + 3)(x - 3)}$, vertical asymptotes are $x=-3$ and $x = 3$. For $f(x)=\frac{119x}{792x+345}$, vertical asymptote is $x=-\frac{345}{792}$. For $f(x)=\frac{612}{816x - 272}$, vertical asymptote is $x=\frac{272}{816}=\frac{17}{51}$. For $f(x)=\frac{237x}{421x-515}$, vertical asymptote is $x=\frac{515}{421}$.

Step2: Analyze horizontal - asymptotes

For a rational function $f(x)=\frac{N(x)}{D(x)}$ where $N(x)=a_nx^n+\cdots+a_0$ and $D(x)=b_mx^m+\cdots + b_0$. If $n=m$, the horizontal asymptote is $y=\frac{a_n}{b_m}$. If $n\lt m$, the horizontal asymptote is $y = 0$. If $n>m$, there is no horizontal asymptote. For $f(x)=\frac{8x^{2}}{x^{2}-9}$, since $n = m=2$, $y=\frac{8}{1}=8$. For $f(x)=\frac{119x}{792x+345}$, since $n = m = 1$, $y=\frac{119}{792}$. For $f(x)=\frac{612}{816x-272}$, since $n=0,m = 1$, $y = 0$. For $f(x)=\frac{237x}{421x-515}$, since $n=m = 1$, $y=\frac{237}{421}$. By observing the graph (not shown here but based on asymptote analysis), we assume the graph has vertical asymptotes $x=\pm3$ and horizontal asymptote $y = 8$.

Answer:

$f(x)=\frac{8x^{2}}{x^{2}-9}$