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

Aug 6, 2017

Use the definition of the absolute value function:

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

Explanation:

Given: $m \left(x\right) = | x + 2 | - 7 \text{ [1]}$

Here is a graph of equation [1]:

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

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

Here is the graph of the two pieces together:

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