Piece wise functions

Question

Hi, how would i create the piece wise function for this : 

 

y=| x+2 | +3

Answer 1

An absolute value function is a special type of function.  When drawn, it represents a V-shape curve.  In this case, we have a right side up V-shape.  These type of functions also have a turning point.  From negative infinity, the curve decreases, then increases at a certain point.  This point is called the vertex.

 

The vertex of this absolute value function is (-2, 3).  Based on this, your piecewise function would be

 

f(x) = { -x ,  x≤-2

              x ,  x>-2 }

 

 

or you can have it as

 

f(x) = { -x ,  x<-2
              x ,  x≥-2 }

 

 

Notice that the signs differ a bit between the two options.

Answer 2

What is needed is treat both part of the absolute value part of the equation as two separate pieces depending if x is greater than or less than -2.  Once this is done then the two parts are combined to represent the different cases.

 

If x ≥ -2 then |x + 2| = x + 2 and y = (x+2) + 3 = x + 5

 

If x < -2 then |x + 2| = -x -2 and y = (-x-2) + 3 = 1 - x

 

Putting those two parts together gives you the piece wise function.

 

y = 1 - x, when x < -2

   = x + 5, when x ≥ -2

Answer 3

You can create a "greatest integer" function:

 

y = | [[x]] + 2 | + 3