2.6 Special Functions

Step functionsGreatest integer functionsPiecewise functions

Step functions: A range of values give a certain outcome.

Your grades are based on a step function.Grade Scale

Letter grades have the following percentage equivalents:

A+ 99-100 B+ 91-92 C+ 83-84 D+ 75-76 F 0- 69

A 96-98 B 88-90 C 80-82 D 72-74

A- 93-95 B- 85-87 C- 77-79 D- 70-71

Greatest Integer Function is a step function

The function is written as

It is not an absolute value. The function rounds down to the last integer.

|][|)( xxf

Find the value of a number in the Greatest Integer function f(x) =[| x |]

f(2.7) = 2 f(0.8) = 0 f(- 3.4) = - 4

It rounds down to the last integer

Find the value

f( 5.8) = f(⅛) = f(- ⅜) =

A step function graph

How to graph a step function; f(x)= [| x |]

Find the values of x = .., -2, -1, 0, 1, 2, ……

f(-2) = -2

f(-1) = - 1

f(0) = 0

f(1) = 1

f(2) = 2

Now lets look at 0.5,1.5, -0.5, -1.5

f(-1.5) = -2 It is the same as f( - 2) = -2f(-0.5) = - 1 f( - 1) = -1f(0.5) = 0 f(0) = 0f(1.5) = 1 f(1) = 1

So between 0 and almost 1 it equal 0f(0.999999999999999999999) = 0

How to show all those number equal 0

A close circle at (0, 0)

and an open circle at (1, 0).

(1, 0)

What happens when x = 1?

How to show all those number equal 0

A close circle at (0, 0)

and an open circle at (1, 0).

(1,1) (2,1)

(1, 0)

What happens when x = 1? It jumps to (1,1)

Is the step only one unit long?

It will be in f(x) = [| x |].

Here is how I graph them.

Find the fill in circles.

Draw line segments ending in a open circle.

The Constant Function

Here f(x) is equal to one number.

f(x) = 3.

Have we seen

this before?

Absolute Value function: f(x) = | x |

Let plot some pointsx f(x)

0 01 1

-1 12 2

-2 2

Absolute Value function: f(x) = | x |

Let plot some pointsx f(x)

0 01 1

-1 - 12 2

-2 - 2Shape V for victory

Lets graph f(x) = - | x – 3|

x - | x – 3| f(x)0 - | 0 – 3| = - | - 3| - 3 (0, - 3)1 - | 1 – 3| = - | - 2| - 2 (1, - 2)2 - | 2 – 3| = - | - 1| - 1 (2, - 1)3 - | 3 – 3| = - | - 0| 0 (3, 0)4 - | 4 – 3| = - | 1 | - 1 (4, - 1)5 - | 5 – 3| = - | 2 | - 2 (5, - 2)

Lets graph f(x) = - | x – 3|

(0, - 3)(1, - 2)(2, - 1)(3, 0)(4, - 1)(5, - 2)

Homework part 1 of section 2.6Homework part 1 of section 2.6

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Piecewise Functions

Graphing different functions over different parts of the graph.

One part tells you what to graph, then where to graph it. What to graph Where to graph

23

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xx

xxxf

Piecewise Functions

2 is where the graph changes.

Less then 2 uses 3x + 2

Greater then 2 uses x - 3

23

223)(

xx

xxxf

We can and should put in a few x into the function

If f(0) we use 3x + 2, then 3(0) + 2 = 2

If f(3) we use x – 3,

then (3) – 3 = 0

The input tell us what function to use.

23

223)(

xx

xxxf

We can and should put in a few x into the function

If we want to find out what f(2) = we use both equations, but leaving an open space on the graph for the point in the function 3x + 2.

Why?

23

223)(

xx

xxxf

We can and should put in a few x into the function

f(2) in 3x + 2; 3(2) + 2 = 8

Graph an open point at (2,8). f(2) in x – 3

(2) – 3 = -1Graphs a filled in point

at (2, -1)

23

223)(

xx

xxxf

Piecewise Functions

So put in an x where the domain changes and one point higher

and lower (2, 8)

(2, -1)

Graph the piecewise function

312

325

22

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xx

x

xx

xg

HomeworkHomework

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