Graph the function. $$ f(x)=\left\{\begin{array}{rr}1-2 x & \text { if } x

Question

Graph the function. $$ f(x)=\left\{\begin{array}{rr}1-2 x & \text { if } x<-1 \ x-2 & \text { if } x \geq-1\end{array}\right. $$

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1. **Identify the pieces of the function**

The function is defined as
   $$
   f(x) =
   \begin{cases}
   1 - 2x, & \text{if } x < -1, \
   x - 2, & \text{if } x \ge -1.
   \end{cases}
   $$

2. **Graphing the first piece: $$\boldsymbol{f(x) = 1 - 2x}$$ for $$\boldsymbol{x < -1}$$**

- **Slope and intercept:** This is a linear function with slope $$-2$$ and $$y$$-intercept $$1$$.
   - **Interval:** It is only valid for $$x < -1$$.
   - **Endpoint behavior:** To see what happens as $$x$$ approaches $$-1$$ from the left, substitute:
     $$
     \lim_{x \to -1^-} f(x) = 1 - 2(-1) = 1+2 = 3.
     $$
     Since $$x = -1$$ is not included in this interval, the point $$(-1, 3)$$ should be shown as an open circle.

3. **Graphing the second piece: $$\boldsymbol{f(x) = x - 2}$$ for $$\boldsymbol{x \ge -1}$$**

- **Slope and intercept:** This is a linear function with slope $$1$$ and $$y$$-intercept $$-2$$.
   - **Interval:** It is defined for $$x \ge -1$$.
   - **Endpoint at $$x = -1$$:** At $$x = -1$$,
     $$
     f(-1) = (-1) - 2 = -3.
     $$
     Hence, the point $$(-1, -3)$$ is included (shown as a solid dot).

4. **Drawing the graph**

- **For $$x < -1$$:** Plot the line $$f(x) = 1 - 2x$$ only for $$x < -1$$. At $$x = -1$$, mark an open circle at $$(-1, 3)$$.
   - **For $$x \ge -1$$:** Plot the line $$f(x) = x - 2$$ starting at the point $$(-1, -3)$$ (solid dot) and continuing to the right.

5. **Observations about the graph**

- There is a jump discontinuity at $$x = -1$$ because the two pieces meet at different $$y$$-values: one is approaching $$3$$ (from the left) and the other starts at $$-3$$ (for $$x \ge -1$$).
   - Both segments are straight lines, but they have different slopes and different $$y$$-intercepts (when extended over their respective domains).

The resulting graph consists of two distinct linear segments with a jump at $$x = -1$$.
To graph the function $$ f(x) $$, plot two lines: 1. For $$ x < -1 $$, draw $$ f(x) = 1 - 2x $$ as a dashed line ending with an open circle at $$ (-1, 3) $$. 2. For $$ x \geq -1 $$, draw $$ f(x) = x - 2 $$ as a solid line starting with a closed circle at $$ (-1, -3) $$. There's a jump at $$ x = -1 $$ where the function changes from $$ 3 $$ to $$ -3 $$.

Graph the function.

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