How do you write #f(x) = 3 - |2x + 3|# as piecewise functions?

1 Answer
georgef
Sep 4, 2016

You have to use the definition of #|t|#, as follows:

#|t|=t#, if #t >= 0#
#|t|=-t#, if #t < 0#

Explanation:

So you have to split the domain of the function depending on whether #2x+3 >=0# or #2x+3 < 0#

If #2x+3 >=0#, that is if #x >= -3/2#, the given function is:
#f(x) = 3 - (2x+3) = -2x#.

Else, if #2x +3 < 0#, that is if #x <-3/2# the given function becomes:
#f(x)=3 - (-2x-3)=6+2x#.

the answer as a piecewise function is then:

#f(x)=-2x#, if #x >= -3/2#
#f(x)=6+2x#, if #x <-3/2#

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