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

1 Answer
Feb 13, 2017

Use the definition of the absolute value function:

|a| ={(a" for "a>=0),(-a" for "a<0):}

and then simplify the domain restrictions.

Explanation:

Applying the definition to |2x+3|:

|2x+3| = {((2x+3)" for "2x+3>=0),(-(2x+3)" for "2x+3<0):}

Simplify the domain restrictions:

|2x+3| = {((2x+3)" for "2x>=-3),(-(2x+3)" for "2x<-3):}

|2x+3| = {((2x+3)" for "x>=-3/2),(-(2x+3)" for "x<-3/2):}" [1]"

Now that we have simplified the domain restrictions, we write f(x) with the right side of equation [1] replacing |2x+3|:

f(x) = {(3 - (2x+3)" for "x>=-3/2),(3 - -(2x+3)" for "x<-3/2):}" [2]"

Use the distributive property to eliminate the ()s:

f(x) = {(3 - 2x-3" for "x>=-3/2),(3 +2x+3" for "x<-3/2):}" [3]"

Equation [3] is the desired piece-wise function.