Lesson  4-­‐1  

Piecewise  Functions  

Recall  the  parent  function  for  absolute  value:  

 

The graphs of both y = x − 2 for x < 3 and y = −2x + 7 for x ≥ 3 are shown on the same coordinate grid below.

How  could  we  rewrite  the  absolute  value  parent  function  as  two  different  functions?  

Lesson  4-­‐1  

A piecewise function is a function that is defined using different rules for the different nonoverlapping intervals of its domain.

 To  evaluate  any  piecewise  function  for  a  specific  x-­‐value:  1. Find  the  interval  of  the  domain  that  contains  that  input    2. Use  the  rule  for  that  interval.  

 Example  #1:  Evaluate  the  Piecewise  function  for  x  =  -­‐1  and  x  =  4.  

       You  try::  Evaluate  the  Piecewise  Function  for  x  =  -­‐2  and  x  =0.    

         

Lesson  4-­‐1  

Example  #2:  Graph  the  function.  Then  identify  the  domain  and  range.  

 

Example #3: Graph  the  function.  Then  identify  the  domain  and  range.

𝑦 = 4𝑥,                    𝑤ℎ𝑒𝑛  𝑥 > 1−𝑥 + 3,      𝑤ℎ𝑒𝑛  𝑥 ≤ 1

Lesson  4-­‐1  

You try: Graph  the  function.  Then  identify  the  domain  and  range.

𝑦 = −𝑥,                    𝑤ℎ𝑒𝑛  𝑥 > 32𝑥 + 1,      𝑤ℎ𝑒𝑛  𝑥 ≤ 3

 Example  #4:  Graph  the  function.  Then  identify  the  domain  and  range.  

𝑦 =−2,                          𝑤ℎ𝑒𝑛  𝑥 < 0  1, 𝑤ℎ𝑒𝑛  0 ≤ 𝑥 ≤ 2  5,                                𝑤ℎ𝑒𝑛  𝑥 > 2

 

 

 A  hole  is  an  open  circle  on  the  graph.A  piecewise  function  that  is  constant  for  each  interval  of  its  domain  is  called  a  step  function.    

Lesson  4-­‐1  

You  try:  Graph  the  function.  Then  identify  the  domain  and  range.  

       Example  #5:  Write  the  piece-­‐wise  function  whose  graph  is  shown.    

         State  the  domain  and  range  of  the  graph.