1. **Problem Statement:** Given the functions $f(x) = \sqrt{x}$ and $g(x) = x^2$, we want to analyze the combined function $$y = f(x) - g(x) = \sqrt{x} - x^2$$ and determine which graph shape it most likely resembles. 2. **Formula and Domain:** The function is $$y = \sqrt{x} - x^2$$ where $x \geq 0$ because $\sqrt{x}$ is defined only for non-negative $x$. 3. **Behavior at Key Points:** - At $x=0$, $$y = \sqrt{0} - 0^2 = 0 - 0 = 0$$ - At $x=1$, $$y = \sqrt{1} - 1^2 = 1 - 1 = 0$$ 4. **Check values between 0 and 1:** - For $x=0.5$, $$y = \sqrt{0.5} - (0.5)^2 = 0.707 - 0.25 = 0.457$$ (positive) 5. **Check values greater than 1:** - For $x=2$, $$y = \sqrt{2} - 2^2 = 1.414 - 4 = -2.586$$ (negative) 6. **Shape Analysis:** - The function starts at 0 at $x=0$. - It rises to a positive value near $x=0.5$. - It returns to 0 at $x=1$. - Then it falls sharply into negative values for $x > 1$. 7. **Conclusion:** The graph starts near $y=0$, rises slightly, then falls downward sharply into negative $y$-values as $x$ increases, matching description (a). **Final answer:** a)