1. **State the problem:** We analyze the rational function $$f(x) = \frac{3x^2 - 13x - 10}{x^2 - 2x - 15}$$ to find points of discontinuity, vertical and horizontal asymptotes, and intercepts. 2. **Factor numerator and denominator:** $$3x^2 - 13x - 10 = (3x + 2)(x - 5)$$ $$x^2 - 2x - 15 = (x - 5)(x + 3)$$ 3. **Simplify the function:** $$f(x) = \frac{(3x + 2)(x - 5)}{(x - 5)(x + 3)}$$ Cancel common factor: $$f(x) = \frac{\cancel{(x - 5)}(3x + 2)}{\cancel{(x - 5)}(x + 3)} = \frac{3x + 2}{x + 3}$$ 4. **Point discontinuity:** Occurs where a factor cancels out in numerator and denominator, here at $$x = 5$$. 5. **Vertical asymptote:** Occurs where denominator is zero and not canceled, here at $$x + 3 = 0 \Rightarrow x = -3$$. 6. **Horizontal asymptote:** Compare degrees of numerator and denominator (both degree 1 after simplification). Horizontal asymptote is ratio of leading coefficients: $$y = \frac{3}{1} = 3$$ 7. **x-intercept:** Set numerator equal to zero: $$3x + 2 = 0 \Rightarrow x = -\frac{2}{3}$$ Coordinate: $$\left(-\frac{2}{3}, 0\right)$$ 8. **y-intercept:** Set $$x=0$$: $$f(0) = \frac{3(0) + 2}{0 + 3} = \frac{2}{3}$$ Coordinate: $$(0, \frac{2}{3})$$ --- **Additional problems:** **Bottom-left equation:** Solve $$\frac{6}{x + 2} = \frac{1}{4} + \frac{x - 7}{x + 2}$$ Multiply both sides by $$4(x+2)$$ to clear denominators: $$4(x+2) \cdot \frac{6}{x+2} = 4(x+2) \cdot \frac{1}{4} + 4(x+2) \cdot \frac{x-7}{x+2}$$ Simplify: $$4 \cancel{(x+2)} \cdot \frac{6}{\cancel{x+2}} = (x+2) + 4 \cancel{(x+2)} \cdot \frac{x-7}{\cancel{x+2}}$$ $$24 = x + 2 + 4(x - 7)$$ Expand: $$24 = x + 2 + 4x - 28$$ $$24 = 5x - 26$$ Add 26: $$24 + 26 = 5x$$ $$50 = 5x$$ Divide: $$x = \frac{50}{5} = 10$$ **Bottom-right inequality:** $$(x + 2)/(x - 4) > 3$$ Rewrite: $$(x + 2)/(x - 4) - 3 > 0$$ Common denominator: $$\frac{x + 2 - 3(x - 4)}{x - 4} > 0$$ Simplify numerator: $$x + 2 - 3x + 12 = -2x + 14$$ Inequality: $$\frac{-2x + 14}{x - 4} > 0$$ Rewrite numerator: $$\frac{-2(x - 7)}{x - 4} > 0$$ Multiply numerator and denominator by -1 (flip inequality): $$\frac{2(x - 7)}{x - 4} < 0$$ Analyze sign: - Numerator zero at $$x=7$$ - Denominator zero at $$x=4$$ Test intervals: - For $$x < 4$$: numerator negative or positive? At $$x=0$$ numerator $$2(0-7) = -14 < 0$$ denominator $$0-4 = -4 < 0$$ fraction positive (negative/negative) - For $$4 < x < 7$$: numerator negative? At $$x=5$$ numerator $$2(5-7) = -4 < 0$$ denominator $$5-4=1 > 0$$ fraction negative - For $$x > 7$$: numerator positive, denominator positive, fraction positive Inequality $$< 0$$ holds for $$4 < x < 7$$ --- **Final answers:** - a. Point discontinuity at $$x = 5$$ - b. Vertical asymptote at $$x = -3$$ - c. Horizontal asymptote at $$y = 3$$ - d. x-intercept at $$\left(-\frac{2}{3}, 0\right)$$ - e. y-intercept at $$(0, \frac{2}{3})$$ - Bottom-left equation solution: $$x = 10$$ - Bottom-right inequality solution: $$4 < x < 7$$