1. **Problem Statement:** We are given the function $f(x) = 2x - 3 \sin(x)$ and asked to analyze it. 2. **Formula and Explanation:** The function combines a linear term $2x$ and a sinusoidal term $-3 \sin(x)$. The linear term causes the function to increase steadily, while the sine term causes oscillations. 3. **Key Properties:** - The sine function oscillates between $-1$ and $1$, so $-3 \sin(x)$ oscillates between $-3$ and $3$. - The linear term $2x$ dominates for large $|x|$, making the function grow positively or negatively without bound. 4. **Evaluate at $x=0$:** $$f(0) = 2 \cdot 0 - 3 \sin(0) = 0 - 0 = 0$$ So the curve passes through the origin $(0,0)$. 5. **Behavior:** - For small $x$, the sine term causes oscillations around the line $y=2x$. - The function is continuous and differentiable everywhere. 6. **Derivative:** $$f'(x) = \frac{d}{dx}(2x) - \frac{d}{dx}(3 \sin(x)) = 2 - 3 \cos(x)$$ This derivative helps find extrema. 7. **Extrema:** Set $f'(x) = 0$: $$2 - 3 \cos(x) = 0 \implies \cos(x) = \frac{2}{3}$$ Solutions for $x$ are where $\cos(x) = \frac{2}{3}$, giving local maxima and minima. **Final summary:** The function $f(x) = 2x - 3 \sin(x)$ is a continuous curve passing through $(0,0)$, oscillating due to the sine term, and increasing overall due to the linear term $2x$.