Question

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

Answer 1

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.