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

Step Functions – functions whose graphs resemble sets of stair steps. Greatest Integer Function – the most notable step function Notation of Greatest Integer Function : x Meaning of Greatest Integer Function : the greatest integer less than or equal to x.

2 2 ? 9 9 ? -3 -3 ? = = = Let’s evaluate some greatest integers… the greatest integer less than or equal to x. -3 = -3 ?

2.2 2 ? 1/2 ? -4.1 -5 ? = = = Let’s evaluate some greatest integers… the greatest integer less than or equal to x. 1/2 = ? -4.1 = -5 ? 1 2 3 -1 -2 -3 -4 -5

Let’s evaluate some greatest integers… 9.1 = ? 9 the greatest integer less than or equal to x. 51/3 = 5 ? -22/9 = -3 ? 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 y .25 .5 .75 1 1.25 1.5 1.75 2 Now that you know how to evaluate greatest integer functions, you can graph them. y = x

x y .25 .5 .75 1 1.25 1.5 1.75 2 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. Example 6-1a

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 Example 6-1b

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. Example 6-1c

Sales The Daily Grind charges $1 Sales The Daily Grind charges $1.25 per pound of meat or any fraction thereof. Draw a graph that represents this situation. Answer: Example 6-1d