Question

express the absolute value function $f(x) = 2|x|$ as a piecewise - defined function.
$f(x)=\begin{cases}2x, & xgeq0\\ - 2, & xlt0end{cases}$

Explanation:

Step1: Recall absolute value definition

The absolute value function \(|x|\) is defined as \(|x|=

$$\begin{cases}x, & x\geq0\\ -x, & x < 0\end{cases}$$

\).

Step2: Apply to \(f(x) = 2|x|\)

Multiply each case of \(|x|\) by 2. For \(x\geq0\), \(2|x|=2\times x = 2x\). For \(x < 0\), \(2|x|=2\times(-x)=-2x\). So the correct piecewise function should be \(f(x)=

$$\begin{cases}2x, & x\geq0\\ -2x, & x < 0\end{cases}$$

\). The given function has a mistake in the \(x < 0\) case (it has \(-2\) instead of \(-2x\)).

Answer:

The correct piecewise - defined function for \(f(x) = 2|x|\) is \(f(x)=

$$\begin{cases}2x, & x\geq0\\ -2x, & x < 0\end{cases}$$

\)