1. **State the problem:** We need to analyze and graph the rational function $$R(x) = \frac{7}{x^2 - 16}$$ and find its domain. 2. **Factor the denominator:** The denominator is a difference of squares: $$x^2 - 16 = (x - 4)(x + 4)$$ So the function in factored form is: $$R(x) = \frac{7}{(x - 4)(x + 4)}$$ 3. **Determine the domain:** The function is undefined where the denominator is zero, so solve: $$(x - 4)(x + 4) = 0 \implies x = 4 \text{ or } x = -4$$ Thus, the domain excludes $x = -4$ and $x = 4$. 4. **Write the domain in interval notation:** $$(-\infty, -4) \cup (-4, 4) \cup (4, \infty)$$ 5. **Identify asymptotes:** - Vertical asymptotes at $x = -4$ and $x = 4$ because the denominator is zero there. - Horizontal asymptote at $y = 0$ because the degree of the denominator (2) is greater than the degree of the numerator (0). 6. **Graph shape:** The graph will have two vertical asymptotes at $x = -4$ and $x = 4$, and the curve approaches $y=0$ as $x \to \pm \infty$. **Final answers:** - Factored form: $$R(x) = \frac{7}{(x - 4)(x + 4)}$$ - Domain: $$(-\infty, -4) \cup (-4, 4) \cup (4, \infty)$$