1. **Problem statement:** Find the number of intersection points of the graphs of the functions for each case. 2. **General approach:** To find the number of intersection points of $G_f$ and $G_g$, solve the equation $f(x) = g(x)$. --- **a) Given:** $$f(x) = \frac{4}{x^2 - 2x + 2}, \quad g(x) = \frac{2}{x + 3}$$ Set equal: $$\frac{4}{x^2 - 2x + 2} = \frac{2}{x + 3}$$ Cross-multiply: $$4(x + 3) = 2(x^2 - 2x + 2)$$ Expand: $$4x + 12 = 2x^2 - 4x + 4$$ Bring all terms to one side: $$0 = 2x^2 - 4x + 4 - 4x - 12$$ $$0 = 2x^2 - 8x - 8$$ Divide both sides by 2: $$0 = \cancel{2}x^2 - \cancel{2}4x - \cancel{2}4$$ $$0 = x^2 - 4x - 4$$ Use the quadratic formula: $$x = \frac{4 \pm \sqrt{(-4)^2 - 4 \cdot 1 \cdot (-4)}}{2} = \frac{4 \pm \sqrt{16 + 16}}{2} = \frac{4 \pm \sqrt{32}}{2}$$ Simplify: $$x = \frac{4 \pm 4\sqrt{2}}{2} = 2 \pm 2\sqrt{2}$$ **Number of solutions:** 2 intersection points. --- **b) Given:** $$f(x) = \frac{3x - 7}{x - 2}, \quad g(x) = 0.6x + 2$$ Set equal: $$\frac{3x - 7}{x - 2} = 0.6x + 2$$ Multiply both sides by $x - 2$: $$(3x - 7) = (0.6x + 2)(x - 2)$$ Expand right side: $$3x - 7 = 0.6x^2 - 1.2x + 2x - 4$$ $$3x - 7 = 0.6x^2 + 0.8x - 4$$ Bring all terms to one side: $$0 = 0.6x^2 + 0.8x - 4 - 3x + 7$$ $$0 = 0.6x^2 - 2.2x + 3$$ Multiply entire equation by 5 to clear decimals: $$0 = 3x^2 - 11x + 15$$ Calculate discriminant: $$\Delta = (-11)^2 - 4 \cdot 3 \cdot 15 = 121 - 180 = -59 < 0$$ Since $\Delta < 0$, no real solutions. **Number of solutions:** 0 intersection points. --- **c) Given:** $$f(x) = \frac{1}{x - 2} + 2, \quad g(x) = \frac{4x - 15}{x - 4}$$ Set equal: $$\frac{1}{x - 2} + 2 = \frac{4x - 15}{x - 4}$$ Multiply both sides by $(x - 2)(x - 4)$ to clear denominators: $$(x - 4) + 2(x - 2)(x - 4) = (4x - 15)(x - 2)$$ Expand left side: $$(x - 4) + 2(x^2 - 6x + 8) = (4x - 15)(x - 2)$$ $$x - 4 + 2x^2 - 12x + 16 = 4x^2 - 8x - 15x + 30$$ Simplify left: $$2x^2 - 11x + 12 = 4x^2 - 23x + 30$$ Bring all terms to one side: $$0 = 4x^2 - 23x + 30 - 2x^2 + 11x - 12$$ $$0 = 2x^2 - 12x + 18$$ Divide by 2: $$0 = x^2 - 6x + 9$$ Factor: $$(x - 3)^2 = 0$$ **Number of solutions:** 1 intersection point (double root at $x=3$).