$$ f(x)=\left\{\begin{array}{ll}x+2 & ext { if } x

Question

$$ f(x)=\left\{\begin{array}{ll}x+2 & 	ext { if } x<2 \ x+1 & 	ext { if } x \geq 2\end{array}ight. $$

UpStudy AI · Accepted Answer

The function $$ f(x) $$ is defined piecewise, meaning it has different expressions based on the value of $$ x $$. Let's break it down:

1. **For $$ x < 2 $$**: The function is defined as $$ f(x) = x + 2 $$.
2. **For $$ x \geq 2 $$**: The function is defined as $$ f(x) = x + 1 $$.

### Analyzing the Function

To understand the behavior of this function, we can analyze it in different intervals:

- **Interval 1**: $$ x < 2 $$
  - Here, $$ f(x) = x + 2 $$.
  
- **Interval 2**: $$ x \geq 2 $$
  - Here, $$ f(x) = x + 1 $$.

### Finding Specific Values

Let's calculate $$ f(x) $$ for a few specific values of $$ x $$:

1. **For $$ x = 1 $$** (which is in the first interval):
   $$
   f(1) = 1 + 2 = 3
   $$

2. **For $$ x = 2 $$** (which is at the boundary):
   $$
   f(2) = 2 + 1 = 3
   $$

3. **For $$ x = 3 $$** (which is in the second interval):
   $$
   f(3) = 3 + 1 = 4
   $$

### Summary of Values

- $$ f(1) = 3 $$
- $$ f(2) = 3 $$
- $$ f(3) = 4 $$

### Graphing the Function

The graph of this piecewise function will consist of two linear segments:

- The line $$ y = x + 2 $$ for $$ x < 2 $$ (which will have a y-intercept at $$ (0, 2) $$ and will slope upwards).
- The line $$ y = x + 1 $$ for $$ x \geq 2 $$ (which will have a y-intercept at $$ (0, 1) $$ and will also slope upwards).

At $$ x = 2 $$, both pieces meet at the point $$ (2, 3) $$.

If you need further analysis or specific calculations, please let me know!
The function $$ f(x) $$ is defined as: - $$ f(x) = x + 2 $$ when $$ x < 2 $$ - $$ f(x) = x + 1 $$ when $$ x \geq 2 $$ For example: - $$ f(1) = 3 $$ - $$ f(2) = 3 $$ - $$ f(3) = 4 $$ The graph consists of two lines meeting at $$ x = 2 $$.

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