identify the following characteristics of the function f(x) = (x² - 9)/(16 - x²): a. vertical asymptotes b…

Explanation:

Step1: Find vertical asymptotes

Set the denominator equal to 0. $$16 - x^{2}=0$$ Factor: $(4 + x)(4 - x)=0$. Solving gives $x = 4$ and $x=-4$.

Step2: Find horizontal asymptotes

Degree of numerator and denominator is 2. Coefficient of $x^{2}$ in numerator is 1 and in denominator is - 1. As $x\to\pm\infty$, $y=\lim_{x\to\pm\infty}\frac{x^{2}-9}{16 - x^{2}}=\lim_{x\to\pm\infty}\frac{1-\frac{9}{x^{2}}}{\frac{16}{x^{2}}-1}=-1$.

Step3: Find y - intercept

Set $x = 0$. Then $f(0)=\frac{0^{2}-9}{16-0^{2}}=-\frac{9}{16}$.

Step4: Find x - intercepts

Set the numerator equal to 0. $x^{2}-9 = 0$, so $(x + 3)(x - 3)=0$. Solving gives $x=3$ and $x=-3$.

Answer:

A. $x = 4,x=-4$ B. $y=-1$ C. $(0,-\frac{9}{16})$ D. $(3,0),(-3,0)$