Presentation on theme: "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."— Presentation transcript:
1 Step functionsThe 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 endThe graph generally resembles a series of steps, hence the name
2 GREATEST INTEGER FUNCTION One prominent example of a step function is the greatest integer functionThis function is written f(x) =The symbol represents the greatest integer LESS THAN OR EQUAL to xFor example, [ 5.2 ] = 5, [2.9] = 2,  = 1Note that integers stay the same, but numbers with a decimal portion are rounded down[-3.4] = -4
3 Step Functions – functions whose graphs resemble sets of stair steps. Greatest Integer Function – the most notable step functionNotation of Greatest Integer Function :xMeaning of Greatest Integer Function :the greatest integer less than orequal to x.
4 2 2 ? 9 9 ? -3 -3 ? = = = Let’s evaluate some greatest integers… the greatest integer less than or equal to x.-3=-3?
5 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?123-1-2-3-4-5
6 Let’s evaluate some greatest integers… 9.1=?9the greatest integer less than or equal to x.51/3=5?-22/9=-3?123-1-2-3-4-5
7 F(x) = [x]Note that we have an open-circle on the right of each step, where the function “jumps up” to the next stepThus 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!
8 xy.25.5.75220.127.116.112Now that you know how to evaluate greatest integer functions, you can graph them.y = x
9 xy.25.5.7518.104.22.1682Now that you know how to evaluate greatest integer functions, you can graph them.y = x - 4
10 Least Integer functions We can also have step functions where you basically round up the numberThese are defined as least integer functions – the function’s output is the least integer that is greater than or equal to the inputSome word problems involve this type of function, such as the ones on the following slides – let’s take a look!
11 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
12 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.42534025517085C(x)xExample 6-1b
13 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
14 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