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

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Answer 1 · 2017-04-28T14:47:21

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

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:

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

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