Chapter 2: Functions and Models

Lesson 7: Step Functions

Mrs. Parziale

Vocabulary

• The greatest integer function is the function f such that for every real number x, f(x) is the greatest integer less than or equal to x.

It is written as

Also known as Floor Function or Rounding Down function

x

Example 1:

• You can fit 12 cans of soda in a box. Make a graph of cans (x) vs. number of complete boxes (y).

• Is this a function? Why / why not?

Greatest Integer Function

• It is written • Also known as the Floor Function or Rounding

Down function. • Round down to the integer to the left on the

number line. • The graph of this is piecewise linear. Each step

is part of a horizontal line.

x

Example 2:

• Evaluate: a) = ______

b) = ______

c) = ______

6.2

3.4

4

• Evaluate: Using your Calculator’s MATH function. (Math Num Int ( )

a) INT (3.7) = ______

b) INT ( ) = ______

c) INT (- ) = ______

Example 2, cont.:

Example 3:

• Graph ( )f x x

X ( )f x x

23 x

12 x

01 x

10 x

21 x

32 x

1 2 3 4 5 6 7 8 9–1–2–3–4–5–6–7–8–9 x

1

2

3

4

5

6

7

8

9

–1

–2

–3

–4

–5

–6

–7

–8

–9

y

What are the domain and range?

Start by examining the graph and then making a table.

• This graph is discontinuous.• The values of x where you lift your pencil off

the paper are the points of discontinuity.

X ( )f x x 23 x

12 x

01 x

10 x

21 x

32 x

1 2 3 4 5 6 7 8 9–1–2–3–4–5–6–7–8–9 x

1

2

3

4

5

6

7

8

9

–1

–2

–3

–4

–5

–6

–7

–8

–9

y

ON A CALCULATOR

• (TI83 or TI83+):• We use the INT function (Math Num int( )

y = int(x)

It looks connected, but it is not.

TI84 does not show this “connection.”

Rounding Up or Ceiling function:

• is the smallest integer greater than or equal to x.

• Note:

( )f x x

x x

3.46 4 3.46 3

3.46 3.46 ( 4) 4

Example 2:

• Evaluate: a) = ______

b) = ______

14.2

5.3

Example 5:

• How many buses (b) are needed to transport (s) students if each bus can hold 44 students?

Example 6:

Suppose it costs $50 to rent a bus in example 5. What will it cost to transport 300 students?

Example 7:

• Graph ( )f x x

1 2 3 4 5 6 7 8 9–1–2–3–4–5–6–7–8–9 x

1

2

3

4

5

6

7

8

9

–1

–2

–3

–4

–5

–6

–7

–8

–9

y

Closure

• How are the greatest integer and the ceiling function different?

• What does the open circle and closed circle mean in either function?

• How would you find the and ?

• Is the graph of the greatest integer a function?

41.89 41.89