Question

rewrite $f(x) = |2x + 4| + 1$ as a piecewise function.
$f(x)=\

$$\begin{cases} \\quad \\text{if } x \\leq \\\\ \\quad \\text{if } x > \\end{cases}$$

$

Explanation:

Step1: Find critical x-value

Set $2x+4=0$
Solve for $x$:
$2x = -4$
$x = -2$

Step2: Define for $x \leq -2$

When $x \leq -2$, $2x+4 \leq 0$, so $|2x+4| = -(2x+4)$.
Simplify:
$-(2x+4) + 1 = -2x -4 + 1 = -2x -3$

Step3: Define for $x > -2$

When $x > -2$, $2x+4 > 0$, so $|2x+4| = 2x+4$.
Simplify:
$2x+4 + 1 = 2x + 5$

Answer:

$f(x)=

$$\begin{cases} -2x - 3 & \text{if } x \leq -2 \\ 2x + 5 & \text{if } x > -2 \end{cases}$$

$