step functions the graph of a step function is not linear – it consists of horizontal line...
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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:
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