Question

How do you write `m(x) = |x+2| - 7` as a piecewise function?

Answer 1

Use the definition of the absolute value function:

`|A| = {(A; A >=0),(-A; A < 0):}`

Explanation:

Given: `m(x) = |x+2| - 7" [1]"`

Here is a graph of equation [1]:

![www.desmos.com/calculator]()

Substitute the definition into equation [1] with `A = x+2`:

`m(x) = {(x+2; x+2 >=0),(-(x+2); x+2 < 0):} - 7" [2]"`

Simplify the domain restrictions:

`m(x) = {(x+2; x >=-2),(-(x+2); x < -2):} - 7" [3]"`

Distribute the minus sign:

`m(x) = {(x+2; x >=-2),(-x-2; x < -2):} - 7" [4]"`

Subtract 7 from each of the pieces:

`m(x) = {(x-5; x >=-2),(-x-9; x < -2):} - 7" [4]"`

Here is the graph of `m(x) = x - 5; x >=-2`

![www.desmos.com/calculator]()

Here is the graph of `m(x) = -x - 9; x <-2`

![www.desmos.com/calculator]()

Here is the graph of the two pieces together:

![www.desmos.com/calculator]()

This shows that equation [4] is the correct answer.