Question

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

Answer 1

Use the fact that `abs(u) = {(u,"if",u >= 0),(-u,"if",u < 0):}`

Explanation:

So

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

We can simplify the expressions

`(x+2)-7 = x-5` and `-(x+2)-7 = -x-9`

and we can solve the inequalities

`(x+2) >= 0` is equivalent to `x >= -2` and
`(x+2) < 0` is equivalent to `x < -2`

So we can write the function:

`m(x) = {(x-5,"if",x >= -2),(-x-9,"if",x < -2):}`.