1. Piecewise Functions

2. Up to now, we’ve been looking at functions represented by a single equation.
In real life, however, functions are represented by a combination of equations, each corresponding to a part of the domain.
These are called piecewise functions.

3. Definition:
Piecewise Function –
a function defined by two or more functions over a specified domain.

4. f(x) = What do they look like? x2 + 1 , x  0 x – 1 , x  0
You can EVALUATE piecewise functions.
You can GRAPH piecewise functions.

5. f(x) = Evaluating Piecewise Functions:
Evaluating piecewise functions is just like evaluating functions that you are already familiar with.
Let’s calculate f(2).
f(x) =
x2 + 1 , x  0
x – 1 , x  0
You are being asked to find y when x = 2.
Since 2  0, you will only substitute into the second part of the function:
f(2) = 2 – 1 = 1

6. f(x) = Let’s calculate f(-2). x2 + 1 , x  0 x – 1 , x  0
You are being asked to find y when x = -2. Since -2  0, you will only substitute into the first part of the function:
f(-2) = (-2)2 + 1 = 5

7. f(x) = Your turn: 2x + 1, x  0 2x + 2, x  0 Evaluate the following:? -3
f(5) = 12 ?
f(1) = 4 ?
f(0) = ? 2

8. f(x) = One more: 3x + 2, x  -2 -x , -2  x  1 x2 – 7x, x  1
Evaluate the following:
f(-2) = ? 2
f(3) =
-12 ?
f(-4) =
-10 ?
f(1) = -6 ?

9.  f(x) = Graphing Piecewise Functions: x2 + 1 , x  0 x – 1 , x  0
Determine the shapes of the graphs.
Parabola and Line
Determine the boundaries of each graph.    
Graph the line where x is greater than or equal to zero.
Graph the parabola where x is less than zero. 

10.   f(x) = Graphing Piecewise Functions: 3x + 2, x  -2 -x, -2  x  1
Determine the shapes of the graphs.
Line, Line, Parabola
Determine the boundaries of each graph.        

11. Special Piecewise Function
A piecewise function that is constant for each interval of its domain, such as the ticket price function, is called a step function. You can describe piecewise functions with a function rule.

12. Graph this function:
This special “staircase” step function is called the greatest integer function which can be written as f(x)=[x].
In this function all non-integers are rounded down to the nearest integer.

13. Piecewise functions are easy to graph with your Calculator.
Select Dot mode
Turn off other functions
Y1=(x+3)(-2 < x)(x < 1)+(3)(x = 1)+(-x + 3)(x > 1)