Step functions

• The graph of a step function is not linear – it consists of horizontal line segments, usually having a closed circle on one end and an open circle on the other end

• The graph generally resembles a series of steps, hence the name

GREATEST INTEGER FUNCTION

• One prominent example of a step function is the greatest integer function

• This function is written f(x) =• The symbol represents the greatest integer

LESS THAN OR EQUAL to x• For example, [ 5.2 ] = 5, [2.9] = 2, [1] = 1• Note that integers stay the same, but numbers

with a decimal portion are rounded down• [-3.4] = -4

x x

Greatest Integer Function – the most notable step function

Step Functions – functions whose graphs resemble sets of stair steps.

Notation of Greatest Integer Function :

xMeaning of Greatest Integer Function : the greatest integer less than or equal to x.

Let’s evaluate some greatest integers…

2 =

9 =

-3

=

2?

9?

-3?

the greatest integer less than or equal to x.

Let’s evaluate some greatest integers…

2.2 =

1/2 =

-4.1 =

2?

0?

-5?

the greatest integer less than or equal to x.

0 1 2 3-1-2-3-4-5

Let’s evaluate some greatest integers…

9.1 =

51/3 =

-22/9 =

9?

5?

-3?

the greatest integer less than or equal to x.

0 1 2 3-1-2-3-4-5

F(x) = [x]

• Note that we have an open-circle on the right of each step, where the function “jumps up” to the next step

• Thus this graph does represent a function – a vertical line will not pass thru more than one point!

• Note also that while the domain is all real numbers, the range is limited to integer values!

x y0 0

.25 0.5 0

.75 01 1

1.25 11.5 1

1.75 12 2

Now that you know how to evaluate greatest integer functions, you can graph them.

y = x

x y0

.25.5

.751

1.251.5

1.752

Now that you know how to evaluate greatest integer functions, you can graph them.

y = x - 4

Least Integer functions

• We can also have step functions where you basically round up the number

• These are defined as least integer functions – the function’s output is the least integer that is greater than or equal to the input

• Some word problems involve this type of function, such as the ones on the following slides – let’s take a look!

Psychology One psychologist charges for counseling sessions at the rate of $85 per hour or any fraction thereof. Draw a graph that represents this solution.

Explore The total charge must be a multiple of $85, so the graph will be the graph of a step function.

Plan If the session is greater than 0 hours, but less than or equal to 1 hour, the cost is $85. If the time is greater than 1 hour, but less than or equal to 2 hours, then the cost is $170, and so on.

Solve Use the pattern of times and costs to make a table, where x is the number of hours of the session and C(x) is the total cost. Then draw the graph.

425

340

255

170

85

C(x)x

Answer:

Examine Since the psychologist rounds any fraction of an hour up to the next whole number, each segment on the graph has a circle at the left endpoint and a dot at the right endpoint.

Sales The Daily Grind charges $1.25 per pound of meat or any fraction thereof. Draw a graph that represents this situation.

Answer: