make a complete graph of the following function. a graphing utility is useful in locating intercepts, local…

Explanation:

Step1: Find x - intercepts

Set $y = f(x)=0$, so $\frac{3x - 4}{x^{2}-1}=0$. Since a fraction is 0 when the numerator is 0 and the denominator is non - zero, $3x-4 = 0$ gives $x=\frac{4}{3}$, and $x^{2}-1=(x + 1)(x - 1)\neq0$ (i.e., $x\neq\pm1$).

Step2: Find y - intercept

Set $x = 0$, then $f(0)=\frac{3\times0 - 4}{0^{2}-1}=4$.

Step3: Find vertical asymptotes

Set the denominator equal to 0: $x^{2}-1=(x + 1)(x - 1)=0$. So the vertical asymptotes are $x=-1$ and $x = 1$.

Step4: Find horizontal asymptote

Since the degree of the numerator is 1 and the degree of the denominator is 2, as $x\to\pm\infty$, $y\to0$. So $y = 0$ is the horizontal asymptote.

Based on the x - intercept at $x=\frac{4}{3}$, y - intercept at $y = 4$, vertical asymptotes $x=\pm1$ and horizontal asymptote $y = 0$, we can analyze the graphs.

Answer:

(Without seeing the actual details of the graphs A, B, C, D, we can't give a specific letter - choice answer. But the above steps help in identifying the correct graph by checking these key features on each of the given graphs.)