Piecewise Functions

• 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.

1 ,13

1 ,12

xifx

xifxxf

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

Evaluate f(x) when x=0, x=2, x=4

2 ,12

2 ,2)(

xifx

xifxxf

•First you have to figure out which equation to use•You NEVER use both

X=0This one fitsInto the top equation

So:0+2=2f(0)=2

X=2This one fits hereSo:2(2) + 1 = 5f(2) = 5

X=4

This one fits hereSo:2(4) + 1 = 9f(4) = 9

Graph:

1 ,3

1 ,)( 2

321

xifx

xifxxf

•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)

12

3, 1

2( )3, 1

x if xf x

x if x

x=1 is the breakingpoint of the graph.

To the left is the topequation.

To the right is thebottom equation.

Graph:

1, 2( )

1, 2

x if xf x

x if x

Point of Discontinuity

Step Functions

43 ,432 ,321 ,210 ,1

)(

xifxifxifxif

xf

43,432,321,210,1

)(

xifxifxifxif

xf

Graph :

01,412,323,234,1

)(

xifxifxifxif

xf

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 nearestinteger.

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.

x x

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].