1. 2.5 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. Piecewise Functions
Piecewise functions are functions that can be represented by more than one equation, with each equation corresponding to a different part of the domain.
Piecewise functions do not always have to be line segments. The “pieces” could be pieces of any type of graph.
This type of function is often used to represent real-life problems like ticket prices.

4. Example x + 1, if x < 1 2, if 1 ≤ x ≤ 3 (x-3)2 + 2, if x > 3

5. One equation gives the value of f(x) when x ≤ 1
And the other when x>1

6. Evaluate f(x) when x=0, x=2, x=4
First you have to figure out which equation to use
You NEVER use both
X=4
X=2
X=0
This one fits
Into the top
equation
So:
0+2=2
f(0)=2
So:
2(4) + 1 = 9
f(4) = 9
This one fits here
So:
2(2) + 1 = 5
f(2) = 5
This one fits here

7. Graph: For all x’s < 1, use the top graph (to the left of 1)
For all x’s ≥ 1, use the bottom graph (to the
right of 1)

8. x=1 is the breaking
point of the graph.
To the left is the top
equation.
To the right is the
bottom equation.

9. Graph:
Point of Discontinuity

10. Step Functions

12. Graph :

14. Special Step Functions
Two particular kinds of step functions are called ceiling functions ( f (x)= and floor functions ( f (x)= ).
In a ceiling function, all nonintegers are rounded up to the nearest
integer.
An example of a ceiling function is when a phone service company charges by the number of minutes used and always rounds up to the nearest integer of minutes.

15. Special Step Functions
In a floor function, all nonintegers are rounded down to the nearest integer.
The way we usually count our age is an example of a floor function since we round our age down to the nearest year and do not add a year to our age until we have passed our birthday.
The floor function is the same thing as the greatest integer function which can be written as f (x)=[x].