Suppose that the function $$ f $$ is defined, for all real numbers, as follows. $$ f(x)=\left\{\begin{array}{cc}-2 x+3 & \text { if } x

Question

Suppose that the function $$ f $$ is defined, for all real numbers, as follows. $$ f(x)=\left\{\begin{array}{cc}-2 x+3 & \text { if } x<2 \ x-3 & \text { if } x \geq 2\end{array}\right. $$ Graph the function $$ f $$. Then determine whether or not the function is continuous

UpStudy AI · Accepted Answer

To analyze the function $$ f $$ defined by

$$
f(x) = \begin{cases}
-2x + 3 & \text{if } x < 2, \
x - 3 & \text{if } x \geq 2,
\end{cases}
$$

let's break down its components, graph the function, and determine its continuity.

## Graphing the Function $$ f $$

The function $$ f $$ is piecewise-defined with two linear segments. Here's how each part behaves:

1. **For $$ x < 2 $$:**
   $$
   f(x) = -2x + 3
   $$
   - **Slope:** -2 (indicates the line is decreasing).
   - **Y-intercept:** 3 (the point where the line crosses the y-axis).
   - **Key Point:** When $$ x = 2 $$, $$ f(2) = -2(2) + 3 = -1 $$. However, since this piece is only for $$ x < 2 $$, the point $$ (2, -1) $$ is not included in this segment.

2. **For $$ x \geq 2 $$:**
   $$
   f(x) = x - 3
   $$
   - **Slope:** 1 (indicates the line is increasing).
   - **Y-intercept:** -3.
   - **Key Point:** At $$ x = 2 $$, $$ f(2) = 2 - 3 = -1 $$. This point **is** included in this segment.

### Plotting the Segments

- **First Segment ($$ x < 2 $$):**
  - Starts from the y-intercept at (0, 3).
  - Passes through the point $$ (2, -1) $$ but does not include it (usually represented with an open circle).

- **Second Segment ($$ x \geq 2 $$):**
  - Starts at $$ (2, -1) $$ (solid dot) since $$ x = 2 $$ is included.
  - Extends infinitely to the right with a slope of 1.

### Visual Representation

While I can't draw the graph here, you can visualize it as follows:

- **Left Line ($$ x < 2 $$):** A straight line descending from the left, passing through (0, 3) and approaching $$ (2, -1) $$ with an open circle at $$ (2, -1) $$.
- **Right Line ($$ x \geq 2 $$):** A straight line ascending from $$ (2, -1) $$ onwards.

![Graph of the function f(x)](https://i.imgur.com/YourImageLink.png)  
*Note: Replace the URL with an actual graph image if available.*

## Determining Continuity

A function is **continuous** at a point $$ x = c $$ if the following three conditions are met:

1. **$$ f(c) $$ is defined.**
2. **The limit $$ \lim_{x \to c} f(x) $$ exists.**
3. **$$ \lim_{x \to c} f(x) = f(c) $$.**

Let's examine the point $$ x = 2 $$, where the definition of $$ f(x) $$ changes.

1. **$$ f(2) $$ is defined:**
   $$
   f(2) = 2 - 3 = -1
   $$

2. **Left-hand limit as $$ x $$ approaches 2:**
   $$
   \lim_{x \to 2^-} f(x) = \lim_{x \to 2^-} (-2x + 3) = -2(2) + 3 = -1
   $$

3. **Right-hand limit as $$ x $$ approaches 2:**
   $$
   \lim_{x \to 2^+} f(x) = \lim_{x \to 2^+} (x - 3) = 2 - 3 = -1
   $$

Since both one-sided limits exist and are equal to each other and to $$ f(2) $$, the function $$ f $$ is **continuous at $$ x = 2 $$**.

Moreover, each piece of $$ f $$ is a linear function, which is continuous everywhere on its domain. Therefore, the entire function $$ f $$ is **continuous for all real numbers**.

## Conclusion

- **Graph:** The function consists of two straight lines:
  - A decreasing line $$ -2x + 3 $$ for $$ x < 2 $$,
  - An increasing line $$ x - 3 $$ for $$ x \geq 2 $$,
  - Both segments meet seamlessly at $$ (2, -1) $$.

- **Continuity:** $$ f(x) $$ is **continuous for all real numbers**.
The function $$ f(x) $$ is continuous for all real numbers.

Suppose that the function \( f \) is defined, for all real numbers, as follows. \[ f(x)=\left\{\begin{array}{cc}-2 x+3 & \text { if } x<2 \\ x-3 & \text { if } x \geq 2\end{array}\right. \] Graph the function \( f \). Then determine whether or not the function is continuous

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