Question

Piecewise linear functions quick check

Answer 1

Piecewise linear functions are functions defined by different linear expressions over different intervals. To perform a quick check on piecewise linear functions, consider the following key points:

### Definition:
A piecewise linear function is typically defined as:
\[
f(x) =
\begin{cases}
m_1x + b_1 & \text{if } x < a_1 \\
m_2x + b_2 & \text{if } a_1 \leq x < a_2 \\
m_3x + b_3 & \text{if } a_2 \leq x < a_3 \\
\vdots & \vdots \\
m_nx + b_n & \text{if } x \geq a_n
\end{cases}
\]
Where \(m_i\) and \(b_i\) are the slope and intercept of each linear segment.

### Steps for Quick Check:

1. **Identify the Intervals**: Determine the intervals defined by the points \(a_1, a_2, \ldots, a_n\).

2. **Check Continuity**: Check if the function is continuous at the transition points \(a_i\):
- For each \(a_i\), ensure that the limit from the left equals the limit from the right.

3. **Evaluate Slopes**: Confirm that the slopes \(m_i\) are correctly defined for the corresponding intervals.

4. **Plot the Function**: If possible, sketch the function using the given linear equations and intervals to visually check for any discontinuities or errors.

5. **Calculate Specific Values**: Substitute specific values of \(x\) from different intervals into their respective linear equations to ensure correct outputs.

6. **Determine Behavior at Infinity**: If applicable, check the behavior of \(f(x)\) as \(x\) approaches \(+\infty\) or \(-\infty\) based on the last piece.

### Example:
Consider the function:

\[
f(x) =
\begin{cases}
2x + 1 & \text{if } x < 0 \\
-x + 1 & \text{if } 0 \leq x < 3 \\
3x - 5 & \text{if } x \geq 3
\end{cases}
\]

- **Identify Intervals**: The intervals are \( (-\infty, 0) \), \([0, 3)\), and \([3, \infty)\).
- **Check Continuity**:
- At \(x=0\): \( \lim_{x \to 0^-} f(x) = 1\) and \(f(0) = 1\), thus continuous at \(x=0\).
- At \(x=3\): \( \lim_{x \to 3^-} f(x) = -2\) and \(f(3) = 4\), thus not continuous at \(x=3\).
- **Evaluate Slopes**: Slopes are \(2, -1, 3\) for appropriate segments.
- **Plot**: Sketch lines for \(x < 0\), \(0 \leq x < 3\), and \(x \geq 3\).
- **Specific Values**: Calculate \(f(-1)\), \(f(1)\), \(f(3)\), and \(f(4)\).
- **Infinity Behavior**: As \(x \to -\infty\), \(f(x) \to -\infty\); as \(x \to \infty\), \(f(x) \to \infty\).

This structured approach allows for a quick check and understanding of piecewise linear functions.