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
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
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Greatest Integer Function – the most notable step function Step Functions – functions whose graphs resemble sets of stair steps. Notation of Greatest Integer Function : Meaning of Greatest Integer Function : the greatest integer less than or equal to x. x
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Let’s evaluate some greatest integers… 2 ? 9 ? -3 ? the greatest integer less than or equal to x. 2 = 9 = -3 =
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Let’s evaluate some greatest integers… 2 ? 0 ? -5 ? the greatest integer less than or equal to x. 2.2 = 1 / 2 = -4.1 = 0 1 2 3 -1 -2 -3 -4 -5
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Let’s evaluate some greatest integers… 9 ? 5 ? -3 ? the greatest integer less than or equal to x. 9.1 = 5 1 / 3 = -2 2 / 9 = 0 1 2 3 -1 -2 -3 -4 -5
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!
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Example 6-1a 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 .
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Example 6-1b 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
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Example 6-1c 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. Answer:
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Example 6-1d 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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