Questions: Sketch a graph of f(x) = -3 if x ≤ -2, x if -2 < x ≤ 2, 3 if x > 2.

Transcript text: Sketch a graph of $f(x)=\left\{\begin{array}{lll}-3 & \text { if } & x \leq-2 \\ x & \text { if } & -22\end{array}\right.$

Solution

Solution Steps

Step 1: Analyze the first piece of the function

The first piece of the function is defined as $f(x) = -3$ if $x \leq -2$. This represents a horizontal line at $y = -3$ for all $x$ values less than or equal to -2. We represent this with a closed circle at the point $(-2, -3)$ and a line extending to the left.

Step 2: Analyze the second piece of the function

The second piece of the function is defined as $f(x) = x$ if $-2 < x \leq 2$. This is a line segment with a slope of 1 passing through the origin. Since $x$ is greater than -2 but not equal to -2, we use an open circle at $(-2, -2)$. Since $x$ is less than or equal to 2, we use a closed circle at $(2, 2)$. We connect the open circle at $(-2, -2)$ and closed circle at $(2, 2)$ by a line segment.

Step 3: Analyze the third piece of the function

The third piece is $f(x) = 3$ if $x > 2$. This is another horizontal line segment at $y=3$ for all values of $x$ greater than 2. We represent this with an open circle at $(2, 3)$ and a line extending horizontally to the right.

Final Answer The graph

The graph consists of three segments:

A horizontal line at $y=-3$ from $x=-\infty$ to $x=-2$ (including $x=-2$).
A line segment connecting (but not including) the point $(-2, -2)$ to the point $(2, 2)$ (including $(2, 2)$).
A horizontal line at $y=3$ from (but not including) $x=2$ to $x=\infty$.
```
   5 |
    |           .
   4 |
    |
   3 |----------------------o
    |           .
   2 |       .
    |     .
   1 |   .
    | .
```
---------+------------------------- -1 | | -2 |o | -3 |---------------------● | -4 | | -5 | | -5 -4 -3 -2 -1 1 2 3 4 5

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