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

1 Answer
Aug 6, 2017

Answer:

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]:

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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#

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

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Here is the graph of the two pieces together:

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