1. **State the problem:** We need to analyze the function $$y = x^{\frac{3}{2}} + x - 1.3$$ and understand its behavior. 2. **Recall the function form:** The function is a sum of a power function $$x^{\frac{3}{2}}$$, a linear term $$x$$, and a constant $$-1.3$$. 3. **Domain:** Since $$x^{\frac{3}{2}} = (\sqrt{x})^3$$, the domain is $$x \geq 0$$ because square root is defined for non-negative $$x$$. 4. **Find intercepts:** - **y-intercept:** At $$x=0$$, $$y = 0 + 0 - 1.3 = -1.3$$. - **x-intercept(s):** Solve $$x^{\frac{3}{2}} + x - 1.3 = 0$$. 5. **Find extrema:** Take derivative: $$y' = \frac{3}{2} x^{\frac{1}{2}} + 1$$ 6. **Set derivative to zero to find critical points:** $$\frac{3}{2} x^{\frac{1}{2}} + 1 = 0$$ 7. **Solve for $$x$$:** $$\frac{3}{2} \sqrt{x} = -1$$ Since $$\sqrt{x} \geq 0$$, the left side is non-negative, but right side is negative, so no real solution. 8. **Conclusion on extrema:** No critical points, so no local maxima or minima. 9. **Summary:** - Domain: $$x \geq 0$$ - y-intercept: $$-1.3$$ - No x-intercepts can be found algebraically easily; numerical methods needed. - No extrema. Final answer: The function $$y = x^{\frac{3}{2}} + x - 1.3$$ is defined for $$x \geq 0$$, has y-intercept at $$-1.3$$, and no local extrema.