Piecewise Graph Aa7F3D

1. **State the problem:** We need to graph the piecewise function: $$f(x)=\begin{cases}-\frac{2}{3}x+1, & x<3 \\ -1, & x\geq 3\end{cases}$$ 2. **Understand the pieces:** - For $x<3$, the function is a line with slope $-\frac{2}{3}$ and y-intercept $1$. - For $x\geq 3$, the function is constant at $-1$. 3. **Calculate the value at $x=3$ for the first piece:** $$f(3) = -\frac{2}{3} \times 3 + 1 = -2 + 1 = -1$$ 4. **Interpret the open and closed circles:** - At $x=3$, the first piece ends with an open circle at $(3,-1)$ because it is defined only for $x<3$. - The second piece starts at $x=3$ with a closed circle at $(3,-1)$ because it is defined for $x\geq 3$. 5. **Graph description:** - Draw the line $y = -\frac{2}{3}x + 1$ for $x<3$ ending with an open circle at $(3,-1)$. - Draw a horizontal ray at $y=-1$ starting at $x=3$ with a closed circle at $(3,-1)$ extending to the right. 6. **Final answer:** The correct graph matches the description: a decreasing line with y-intercept $1$ ending with an open circle at $(3,-1)$, and a horizontal ray at $y=-1$ for $x\geq 3$ with a closed circle at $(3,-1)$.