# Piecewise Functions and Greatest Integer Function

```Alg 2 BC – U2
Day 3 – Piecewise Functions
Remember: A piecewise function is composed of pieces of other functions.
 x  5,

For example: f ( x)  2 x  1,
2 x  9,

x  2
2 x  2
x2
Graph it below and find f(-10), f(0) and f(10).
Sometimes, piecewise functions are continuous like above, where all of the pieces are connected.
Sometimes a piecewise function consists of pieces that are not connected.
3,

For example: f ( x)   x  1,
 4  x,

x0
0 x2
x2
Graph it below and find f(0) and f(2).
Try to write a piecewise function for the graph below!
A special type of piecewise function is the Greatest Integer Function. It is denoted as f(x) = x .
To see what this graph looks like, graph: y = int(x)
Can you tell, by looking at the graph, what x does?
Using the TABLE function:
Choose 2nd TBL SET: Experiment to see if you can determine what each of the following mean: (To see the
table, you need to choose TABLE after doing TBL SET.)
TblStart
Tbl
Now use the TABLE function to fill in the chart below for f(x) = x
X
0
0.1
0.2
0.5
0.9
0.999
1
Y
We can graph any greatest integer function by simply making a table of values. We could also use
transformations, but we’ll get to that in a few days.
Example:
y  x 2
y 3 x
Extra info: Graphing a Piecewise Function on the Calculator:
 x  5,

For f ( x)  2 x  1,
2 x  9,

x  2
2  x  2,
x2
y1  ( x  5) / ( x  2)
the notation would be y 2  (2 x  1) / ( 2  x and x  2) .
y3  (2 x  9) / ( x  2)
The greater than and less than symbols can be found under the TEST menu (2nd MATH).
3,

Try: f ( x)   x  1,
 4  x,

x0
0 x2
x2
Homework Worksheet: Graph each piecewise function by hand:
2 x  3, x  1

1. f ( x)  1,
 2  x 1
 x  1,
x  2

3x  4, x  2

2. f ( x)   1
x  3, x  2

2
Graph each function by hand by making a table of values:
3. f ( x)  x  2
5. Write a piecewise function for this graph:
4. f ( x)  2 x 1
```