Sketch a graph of $$ f(x)=\left\{\begin{array}{ll}-3 & ext { if } x \leq-1 \ x & ext { if }-12\end{array} ight. $$

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Sketch a graph of $$ f(x)=\left\{\begin{array}{ll}-3 & 	ext { if } x \leq-1 \ x & 	ext { if }-12\end{array}ight. $$

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Certainly! Let's break down how to sketch the graph of the piecewise function: $$ f(x) = \begin{cases} -3 & \text{if } x \leq -1 \ x & \text{if } -1 < x \leq 2 \ 1 & \text{if } x > 2 \end{cases} $$ ### Step-by-Step Guide to Sketching the Graph 1. **Identify the Different Regions:** - **Region 1:** $$ x \leq -1 $$ where $$ f(x) = -3 $$. - **Region 2:** $$ -1 < x \leq 2 $$ where $$ f(x) = x $$. - **Region 3:** $$ x > 2 $$ where $$ f(x) = 1 $$. 2. **Plot Each Region Separately:** - **Region 1 ($$ x \leq -1 $$):** - This is a horizontal line at $$ y = -3 $$. - **Endpoint at $$ x = -1 $$:** - Since $$ x = -1 $$ is included in this region, place a **closed circle** at $$ (-1, -3) $$. - **Region 2 ($$ -1 < x \leq 2 $$):** - This is the line $$ y = x $$, which is a straight line with a slope of 1 passing through the origin. - **Start Point just greater than $$ x = -1 $$:** - At $$ x = -1 $$, $$ f(x) $$ approaches $$ -1 $$ from the right, but $$ x = -1 $$ is not included in this region. Place an **open circle** at $$ (-1, -1) $$ to indicate that $$ x = -1 $$ is not part of this segment. - **Endpoint at $$ x = 2 $$:** - $$ x = 2 $$ is included, so place a **closed circle** at $$ (2, 2) $$. - **Region 3 ($$ x > 2 $$):** - This is a horizontal line at $$ y = 1 $$. - **Start Point just greater than $$ x = 2 $$:** - At $$ x = 2 $$, the previous region reaches $$ y = 2 $$, but $$ x = 2 $$ is not included in this region. Place an **open circle** at $$ (2, 1) $$. 3. **Combine All Regions:** - **From $$ x = -\infty $$ to $$ x = -1 $$:** Draw a horizontal line at $$ y = -3 $$ with a closed circle at $$ (-1, -3) $$. - **From $$ x = -1 $$ to $$ x = 2 $$:** Draw the line $$ y = x $$ starting just above $$ x = -1 $$ (with an open circle at $$ (-1, -1) $$) and ending at $$ (2, 2) $$ with a closed circle. - **From $$ x = 2 $$ to $$ \infty $$:** Draw a horizontal line at $$ y = 1 $$ starting just after $$ x = 2 $$ (with an open circle at $$ (2, 1) $$). ### Visual Representation While I can't provide a visual graph here, here's a textual representation to help you visualize: ``` y | 3 | 2 | ●-------- 1 | ○--------- 0 | ○ -1|●------------------------- |____________________________ x -1 0 2 ``` - **●** represents a closed circle. - **○** represents an open circle. - The horizontal lines are at $$ y = -3 $$ and $$ y = 1 $$. - The diagonal line represents $$ y = x $$ between $$ x = -1 $$ and $$ x = 2 $$. ### Key Points to Remember - **Closed Circle (●):** The point is included in the graph. - **Open Circle (○):** The point is not included in the graph. - **Continuity:** Check for any jumps or gaps in the graph at the boundary points ($$ x = -1 $$ and $$ x = 2 $$). By following these steps, you should be able to accurately sketch the graph of the given piecewise function. To sketch the graph of $$ f(x) $$, plot a horizontal line at $$ y = -3 $$ from $$ x = -\infty $$ to $$ x = -1 $$ with a closed circle at $$ (-1, -3) $$. Then, draw the line $$ y = x $$ from just above $$ x = -1 $$ to $$ x = 2 $$ with an open circle at $$ (-1, -1) $$ and a closed circle at $$ (2, 2) $$. Finally, draw a horizontal line at $$ y = 1 $$ from just after $$ x = 2 $$ with an open circle at $$ (2, 1) $$.

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