3. 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

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

5. 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

6. 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

8. Now that you know how to evaluate greatest integer functions, you can graph them. x y 0 0 .25 0 .5 0 .75 0 1 1 1.25 1 1.5 1 1.75 1 2 2 y = x

9. Now that you know how to evaluate greatest integer functions, you can graph them. y = x - 4 x y 0 .25 .5 .75 1 1.25 1.5 1.75 2

11. 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 .

12. 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

13. 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:

14. 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: