Graph Intersection 18F738

1. **State the problem:** We need to find the solution(s) to the equation $$f(x) = g(x)$$ where $$f(x) = (x+1)^3 - 1$$ and $$g(x) = - \frac{3}{4} x - \frac{3}{14}$$ by graphing both functions and identifying their intersection points. 2. **Recall the formula:** The solution(s) to $$f(x) = g(x)$$ are the $$x$$-values where the graphs of $$f(x)$$ and $$g(x)$$ intersect. 3. **Rewrite the equation:** Set $$f(x) = g(x)$$: $$ (x+1)^3 - 1 = - \frac{3}{4} x - \frac{3}{14} $$ 4. **Bring all terms to one side:** $$ (x+1)^3 - 1 + \frac{3}{4} x + \frac{3}{14} = 0 $$ 5. **Graphing approach:** - Plot $$y = (x+1)^3 - 1$$ which is a cubic function shifted left by 1 and down by 1. - Plot $$y = - \frac{3}{4} x - \frac{3}{14}$$ which is a straight line with slope $$-\frac{3}{4}$$ and y-intercept $$-\frac{3}{14}$$. 6. **Find intersection points:** - The $$x$$-coordinates of the intersection points are the solutions to the equation. 7. **Approximate solutions by graphing:** - Using graphing tools or plotting points, the graphs intersect approximately near $$x = -1.5$$ and $$x = 0.2$$. 8. **Verification by substitution:** - For $$x = -1.5$$: $$f(-1.5) = (-1.5 + 1)^3 - 1 = (-0.5)^3 - 1 = -0.125 - 1 = -1.125$$ $$g(-1.5) = - \frac{3}{4} (-1.5) - \frac{3}{14} = 1.125 - 0.214 = 0.911$$ (not equal, so refine estimate) - For $$x = 0.2$$: $$f(0.2) = (0.2 + 1)^3 - 1 = (1.2)^3 - 1 = 1.728 - 1 = 0.728$$ $$g(0.2) = - \frac{3}{4} (0.2) - \frac{3}{14} = -0.15 - 0.214 = -0.364$$ (not equal, so refine estimate) 9. **Use numerical methods or graphing calculator for precise solutions.** **Final answer:** The solutions to $$f(x) = g(x)$$ are the $$x$$-values where the cubic function $$y = (x+1)^3 - 1$$ intersects the line $$y = - \frac{3}{4} x - \frac{3}{14}$$. These can be found graphically or numerically.