Piecewise Function Graph 53Ff73

1. **State the problem:** We have a piecewise function defined as: $$ f(x) = \begin{cases} 5, & x < -2 \\ 3, & -2 \leq x < 0 \\ 0, & 0 \leq x < 2 \\ -3, & x \geq 2 \end{cases} $$ We need to identify the correct graph representing this function. 2. **Understand the piecewise function:** - For $x < -2$, $f(x) = 5$ (constant value 5) - For $-2 \leq x < 0$, $f(x) = 3$ (constant value 3) - For $0 \leq x < 2$, $f(x) = 0$ (constant value 0) - For $x \geq 2$, $f(x) = -3$ (constant value -3) 3. **Important rules for graphing piecewise functions:** - Closed circle at points where the function is defined (inclusive). - Open circle at points where the function is not defined (exclusive). 4. **Analyze each interval and endpoint:** - At $x = -2$, since $f(x) = 3$ for $-2 \leq x < 0$, the point $(-2, 3)$ has a closed circle. - At $x = 0$, since $f(x) = 0$ for $0 \leq x < 2$, the point $(0, 0)$ has a closed circle. - At $x = 2$, since $f(x) = -3$ for $x \geq 2$, the point $(2, -3)$ has a closed circle. 5. **Check the values just before the closed circles:** - Just before $x = -2$, $f(x) = 5$ for $x < -2$, so at $x = -2$ from the left, the value is 5 but not included (open circle at $(-2, 5)$). - Just before $x = 0$, $f(x) = 3$ for $-2 \leq x < 0$, so at $x = 0$ from the left, the value is 3 but not included (open circle at $(0, 3)$). - Just before $x = 2$, $f(x) = 0$ for $0 \leq x < 2$, so at $x = 2$ from the left, the value is 0 but not included (open circle at $(2, 0)$). 6. **Summary of points and circles:** - $(-2, 5)$ open circle, $(-2, 3)$ closed circle - $(0, 3)$ open circle, $(0, 0)$ closed circle - $(2, 0)$ open circle, $(2, -3)$ closed circle 7. **Conclusion:** The correct graph must have these points and circles matching the above description. **Final answer:** The graph with closed circles at $(-2, 3)$, $(0, 0)$, and $(2, -3)$ and open circles at $(-2, 5)$, $(0, 3)$, and $(2, 0)$ correctly represents the given piecewise function.