The Greatest Integer Function

• y = [x] means “the greatest integer not greater than x”

• For example, [2.4], means “the greatest integer not greater than 2.4.” So, [2.4] = 2.

• Here are some other examples:• [3] = 3

• [-2.2] = -3

• [5.8] = 5

Greatest Integer Function, continued

• Notice that your answer will always be an integer.

• Because we can input any number in for x, but only get integers as our output y, we need to look at this graph because it probably looks VERY different from all of the others we have studied so far.

Let’s graph y = [x] using a table of values:

x y

-.5 -1

-.1 -1

0 0

0.4 0

0.8 0

1 1

1.2 1

1.8 1

2 2

2.3 2

3 3

3.2 3

3.7 3

Because this graph looks like a staircase, we also call it a “step” function.

Notice the open circle at the end of each step. These occur when we “jump” to the next integer.

The domain of y = [x] would be all real numbers, because we can substitute any number in for x and get a result.

The range of y = [x] would be the set of integers, because even though we can substitute decimals or fractions, our answer will always be an integer.

Another Example: y = 2[x – 1]x y

0 -2

0.4 -2

0.8 -2

1 0

1.2 0

1.8 0

2 2

2.3 2

3 4

3.2 4

y = 2[x – 1] continued Notice the steps are

further apart this time: there are 2 spaces between each one.

Also, the step at x = 0 has shifted to the right 1 unit.

The domain of y = 2[x –1] is all real numbers.

The range of y = 2[x – 1] is all even integers, because we will only get even integers as outputs.

Another Example: y = 0.5[x] + 2x y

0 2

0.4 2

0.8 2

1 2.5

1.2 2.5

1.8 2.5

2 3

2.3 3

3 3.5

3.2 3.5

y = 0.5[x] + 2 continued Notice the steps are

closer together this time: there is a half of a space between each one.

Also, the step at x = 0 has shifted up 2 units.

The domain of y = 0.5[x] + 2 is all real numbers.

The range of y = 0.5[x] + 2 is the set of numbers {…, -1.5, -1, -0.5, 0, 0.5, …}, because these are the numbers we will get as outputs.

Transformations of y = [x]

•General form: y = a[x – h] + k

• There will be a units between steps.

• The graph will shift h units right for [x – h] and h units left for [x + h].

• The graph will shift up k units for +k and down k units for –k.

Graphing Step Functions on the Calculator Go to y = as usual. Input your function.

You can find the greatest integer function by going to MATH, moving over to NUM, and choosing #5 int(.

For example, y = [x] + 2 would be put in as y = int(x) + 2.

Then, graph.

Graphing Step Functions on the Calculator continued

• Notice that it looks like the steps are connected, which we know is not the case. This is because of your calculator’s mode.

• Press MODE, and change CONNECTED to DOT.

• Then, graph again. There are our steps!

Step Functions in the Real World

• Cell phone plans: You pay one price for a specific number of minutes. If you want more minutes, you pay more money. Think of each step as a plan.

• The Post Office: You pay postage to mail things based on their weight. One stamp allows you to mail something that weighs up to17 ounces. Past the 17 ounces, you must add another stamp, which means two stamps allows you to mail up to 34 ounces. Think of each step as representing the weight you can mail per stamp.