1. **State the problem:** We need to analyze the function $$f(x) = \frac{2x^2 - 3x + 1}{x^3 + 1}$$ and understand its behavior. 2. **Recall the formula and rules:** The function is a rational function, which is a ratio of two polynomials. Important points include: - Domain restrictions where the denominator is zero. - Simplification if possible. - Finding intercepts by setting numerator or denominator to zero. 3. **Find domain restrictions:** The denominator is $$x^3 + 1$$. Set it to zero: $$x^3 + 1 = 0 \implies x^3 = -1 \implies x = -1$$ So, $$x = -1$$ is not in the domain. 4. **Factor numerator and denominator:** - Numerator: $$2x^2 - 3x + 1$$. Factor: $$2x^2 - 3x + 1 = (2x - 1)(x - 1)$$ - Denominator: $$x^3 + 1$$ is a sum of cubes: $$x^3 + 1 = (x + 1)(x^2 - x + 1)$$ 5. **Simplify the function:** $$f(x) = \frac{(2x - 1)(x - 1)}{(x + 1)(x^2 - x + 1)}$$ No common factors to cancel. 6. **Find intercepts:** - **x-intercepts:** Set numerator to zero: $$2x - 1 = 0 \implies x = \frac{1}{2}$$ $$x - 1 = 0 \implies x = 1$$ - **y-intercept:** Set $$x=0$$: $$f(0) = \frac{2(0)^2 - 3(0) + 1}{0^3 + 1} = \frac{1}{1} = 1$$ 7. **Summary:** - Domain: all real numbers except $$x = -1$$ - x-intercepts at $$x = \frac{1}{2}$$ and $$x = 1$$ - y-intercept at $$y = 1$$ - Vertical asymptote at $$x = -1$$ due to denominator zero. This analysis helps understand the function's behavior and key points.