Piecewise Functions: Sketching and Interpretation

Q2. Sketch the graph of the following piecewise function:

$u(x) = \begin{cases} 2x - 5, & \text{if } x \leq 0 \\ -x - 5, & \text{if } x > 0 \end{cases}$

Background

Topic: Piecewise Functions

This question tests your understanding of how to interpret and graph piecewise-defined functions. Piecewise functions are defined by different expressions depending on the value of the input variable, and their graphs often have distinct segments or behaviors at certain points.

Key Terms and Formulas:

• Piecewise Function: A function defined by multiple sub-functions, each applying to a certain interval of the domain.

• Linear Function: A function of the form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

• Graphing: Plot each segment according to its formula and domain, paying attention to endpoints and possible discontinuities.

Step-by-Step Guidance

1. Identify the two cases for $u(x)$: For $x \leq 0$, use $u(x) = 2x - 5$. For $x > 0$, use $u(x) = -x - 5$.

2. For $x \leq 0$, plot the line $y = 2x - 5$. This is a straight line with slope 2 and y-intercept -5. Make sure to only plot this for $x \leq 0$.

3. For $x > 0$, plot the line $y = -x - 5$. This is a straight line with slope -1 and y-intercept -5. Only plot this for $x > 0$.

4. Check the value at $x = 0$ for both pieces: $u(0) = 2 \cdot 0 - 5 = -5$ (for $x \leq 0$). For $x > 0$, the function does not include $x = 0$, so the graph may have a sharp corner or change in slope at $x = 0$.

5. Look for any discontinuity or sharp change at $x = 0$ by comparing the left and right limits at $x = 0$.

Try solving on your own before revealing the answer!

Final Answer:

The correct graph is option (a), which shows the two linear segments meeting at $x = 0$ with a sharp corner at $(-5, 0)$.

This matches the behavior of the piecewise function, with the left segment ($x \leq 0$) having slope 2 and the right segment ($x > 0$) having slope -1.