1. **Stating the problem:** We want to analyze the function $$f(x) = \frac{\sqrt{x} + \sin(x)}{\cos(x)}$$ and understand its behavior, including points where it is defined and its graph characteristics. 2. **Formula and important rules:** The function is a quotient of two expressions: numerator $$\sqrt{x} + \sin(x)$$ and denominator $$\cos(x)$$. The function is defined only where $$\cos(x) \neq 0$$ and $$x \geq 0$$ (since $$\sqrt{x}$$ requires non-negative $$x$$). 3. **Domain considerations:** - $$\sqrt{x}$$ is defined for $$x \geq 0$$. - $$\cos(x) = 0$$ at $$x = \frac{\pi}{2} + k\pi$$ for integers $$k$$, so these points are excluded from the domain. 4. **Intermediate work:** - The function cannot be simplified further algebraically. - We note the points mentioned: (0,0), (1,1), and (2,0) lie on the graph. 5. **Explanation:** - At $$x=0$$, $$f(0) = \frac{\sqrt{0} + \sin(0)}{\cos(0)} = \frac{0 + 0}{1} = 0$$. - At $$x=1$$, $$f(1) = \frac{\sqrt{1} + \sin(1)}{\cos(1)} = \frac{1 + \sin(1)}{\cos(1)}$$ which numerically is approximately 1. - At $$x=2$$, $$f(2) = \frac{\sqrt{2} + \sin(2)}{\cos(2)}$$ which is approximately 0. This matches the points given. 6. **Summary:** The function $$f(x) = \frac{\sqrt{x} + \sin(x)}{\cos(x)}$$ is defined for $$x \geq 0$$ except where $$\cos(x) = 0$$. It passes through the points (0,0), (1,1), and (2,0) as described.