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.