# How do you write  y=|x+1|-4 as a piecewise function?

Jan 16, 2018

$\textcolor{b l u e}{| x + 1 | - 4 = \left\{x + 1 , \mathmr{if} x \ge \left(- 1\right)\right\}}$

$\textcolor{b l u e}{| x + 1 | - 4 = \left\{- x - 1 , \mathmr{if} x < \left(- 1\right)\right\}}$

#### Explanation:

Given:

$\textcolor{red}{y = f \left(x\right) = | x + 1 | - 4}$

We draw the graph of this function first

$\textcolor{g r e e n}{S t e p .1}$

We find the boundary line first.

Later, once we find the "Piece-wise Functions", we can graph those as well and compare the graphs.

We can accomplish this process by setting what is inside absolute value to ZERO, and then solving for $\textcolor{red}{x}$.

So, when

$x + 1 = 0$

we get

$\textcolor{b l u e}{x = \left(- 1\right)}$

$\textcolor{g r e e n}{S t e p .2}$

When $\left(x + 1\right)$ is Positive, we just consider the expression as it is,

but if $\left(x + 1\right)$ is Negative, we must negate the whole expression

$\textcolor{g r e e n}{S t e p .3}$

Hence,

our required Piece-wise Functions are

$\textcolor{b l u e}{| x + 1 | - 4 = \left\{x + 1 , \mathmr{if} x \ge \left(- 1\right)\right\}}$

$\textcolor{b l u e}{| x + 1 | - 4 = \left\{- x - 1 , \mathmr{if} x < \left(- 1\right)\right\}}$

We will graph the Piece-wise Functions below:

Hope this helps.