Question
\( f(x)=\left\{\begin{array}{ll}x+2 & \text { if } x<2 \\ x+1 & \text { if } x \geq 2\end{array}\right. \)
Ask by Powell Sullivan. in the United States
Mar 14,2025
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Answer
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 \( x = 2 \).
Solution
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!
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Simplify this solution Bonus Knowledge
Did you know that piecewise functions like this one are a fantastic way to model real-life scenarios where different conditions apply? For instance, a person might get charged a different rate for bus fare depending on whether they are traveling on a weekday or a weekend. So, the function could represent their fare depending on the day! When working with piecewise functions, one common mistake is forgetting to check the boundary points where the definition changes. In this case, at \( x = 2 \), the function switches from \( x + 2 \) to \( x + 1 \). It's essential to verify which part of the function applies and calculate the output accurately to avoid errors!
