# How do you write f(x) = 3 - |2x + 3| as piecewise functions?

Sep 4, 2016

You have to use the definition of $| t |$, as follows:

$| t | = t$, if $t \ge 0$
$| t | = - t$, if $t < 0$

#### Explanation:

So you have to split the domain of the function depending on whether $2 x + 3 \ge 0$ or $2 x + 3 < 0$

If $2 x + 3 \ge 0$, that is if $x \ge - \frac{3}{2}$, the given function is:
$f \left(x\right) = 3 - \left(2 x + 3\right) = - 2 x$.

Else, if $2 x + 3 < 0$, that is if $x < - \frac{3}{2}$ the given function becomes:
$f \left(x\right) = 3 - \left(- 2 x - 3\right) = 6 + 2 x$.

the answer as a piecewise function is then:

$f \left(x\right) = - 2 x$, if $x \ge - \frac{3}{2}$
$f \left(x\right) = 6 + 2 x$, if $x < - \frac{3}{2}$