1. **State the problem:** We need to analyze and graph the rational function $$R(x) = \frac{7}{x^2 - 16}$$ including its domain, intercepts, and simplification. 2. **Find the domain:** The domain excludes values where the denominator is zero. Set denominator equal to zero: $$x^2 - 16 = 0$$ $$x^2 = 16$$ $$x = \pm 4$$ So, the domain is all real numbers except $$x = -4$$ and $$x = 4$$. In interval notation, the domain is: $$(-\infty, -4) \cup (-4, 4) \cup (4, \infty)$$ 3. **Simplify the function:** Factor the denominator: $$x^2 - 16 = (x - 4)(x + 4)$$ Since numerator 7 has no common factors with denominator, the function is already in lowest terms. Answer: $$R(x) = \frac{7}{(x - 4)(x + 4)}$$ and it is already in lowest terms. 4. **Find intercepts:** - **x-intercepts:** Set numerator equal to zero: $$7 = 0$$ No solution, so no x-intercepts. - **y-intercept:** Evaluate $$R(0)$$: $$R(0) = \frac{7}{0^2 - 16} = \frac{7}{-16} = -\frac{7}{16}$$ So, y-intercept is $$-\frac{7}{16}$$. 5. **Summary:** - Domain: $$(-\infty, -4) \cup (-4, 4) \cup (4, \infty)$$ - Lowest terms: $$R(x) = \frac{7}{(x - 4)(x + 4)}$$ - Intercepts: No x-intercepts, y-intercept at $$-\frac{7}{16}$$ 6. **Graph shape:** - Vertical asymptotes at $$x = -4$$ and $$x = 4$$ - Horizontal asymptote at $$y = 0$$ (since degree numerator < degree denominator) This completes the analysis of the function.