2.1.5 Piecewise Functions and Absolute Value¶

Expressing absolute value functions as piecewise-defined functions and analyzing their behavior on different intervals.

定义¶

An absolute value function can be expressed as a piecewise-defined function by analyzing the expression inside the absolute value. For a function $f(x) = |g(x)|$, we identify where $g(x) = 0$ (critical points) and determine the sign of $g(x)$ on each interval. The piecewise form is:

$$f(x) = |g(x)| = \begin{cases} g(x) & \text{if } g(x) \geq 0 \\ -g(x) & \text{if } g(x) < 0 \end{cases}$$ \nThis decomposition allows us to analyze the function's behavior separately on different intervals, determine continuity and differentiability, and sketch accurate graphs. The critical points where the expression changes form are typically where the derivative is undefined or where the function has a "corner" or sharp turn in its graph.

核心公式¶

$f(x) = |g(x)| = \begin{cases} g(x) & \text{if } g(x) \geq 0 \\ -g(x) & \text{if } g(x) < 0 \end{cases}$
$|x - a| = \begin{cases} x - a & \text{if } x \geq a \\ -(x - a) & \text{if } x < a \end{cases}$
$|ax + b| = \begin{cases} ax + b & \text{if } x \geq -\frac{b}{a} \\ -(ax + b) & \text{if } x < -\frac{b}{a} \end{cases}$ (for $a > 0$)
$\frac{d}{dx}|g(x)| = \frac{g(x)}{|g(x)|} \cdot g'(x)$ for $g(x) \neq 0$
$\int |g(x)| \, dx = \int_{\text{each interval}} \pm g(x) \, dx$ (evaluated separately on intervals where $g(x)$ has constant sign)

易错点¶

⚠️ Forgetting to check the sign of the expression at critical points: Students often incorrectly determine which piece of the piecewise function applies on each interval, leading to wrong function definitions and incorrect graphs.
⚠️ Assuming the absolute value function is differentiable everywhere: At points where the expression inside the absolute value equals zero, the function typically has a corner and is not differentiable there. Students often forget to exclude these points when finding derivatives.
⚠️ Incorrectly handling the derivative at critical points: When computing $\frac{d}{dx}|g(x)|$, students may forget that the derivative is undefined where $g(x) = 0$, or they may apply the power rule without considering the piecewise nature of the function.
⚠️ Misidentifying the domain where each piece applies: Students sometimes reverse the inequality signs or use $>$ instead of $\geq$ (or vice versa), causing discontinuities or overlaps in their piecewise definition.