STEP FUNCTIONS 3.9

INTRODUCTION In 2007, the U.S. postage rate for first class flats was $0.70 for the first ounce plus $0.17 for each additional ounce or part of an ounce. First- class mail rates are given in the table and graphed below. Notice that the phrase “up to and including the given weight” means that the weight is rounded up to the nearest ounce. For instance, an envelope weighing 4.4 ounces is charged at the 5-ounce rate.

INTRODUCTION (CONTINUED) What is the significance of the opened and closed circles? Because the cost is rounded up, the left end of each segment is not included on the graph and the right end of each segment is included. Is the graph a function? Because no single weight has two costs, the graph is a function. What is the domain? 0 < weight ≤ 13 What is the range? The set of costs {$0.70, $0.87, $1.04, …, $2.74}

FLOOR AND CEILING FUNCTIONS The graph is a specific type of piecewise linear function called a step-function. There are two types of step functions floor function (greatest-integer function, rounding-down function) ceiling function (rounding-up function)

EXAMPLES Evaluate The answer is the greatest integer less than or equal to 57/8. Evaluate The answer is the least integer greater than or equal to π. Evaluate

GRAPHING STEP-FUNCTIONS Graph Make a table of values to see the pattern. x -3 ≤ x < ≤ x < -1

APPLICATIONS OF STEP FUNCTIONS The floor or ceiling function is appropriate when function values must be integers and other formulas would give non- integer values. Example: In March 2008, New York City taxi rates were an initial fee of $2.50 plus $0.40 for each 1/5 mile traveled. Write a formula for T(m), the charge for a trip of m miles. Multiply the number of miles, m, by 5 to determine the number of 1/5 miles traveled to get 5m. Since 5m may not be a whole number (and it needs to be before the company will charge), we can use the greatest-integer function to change it to an integer. An equation for the function would be T(m) = What is the charge for an 8.75 mile trip in a New York City taxi? T(8.75) =

EXAMPLE 2 Users of pre-paid calling cards are billed in 1-minute increments. This means that customers are billed for a full minute when any part of a minute is used. The Call-Me-Often Phone Card Company charges $0.03 per minute with a 1-minute billing increment. Write a formula for P(m), the charge for a call of m minutes. What is the charge for a 5-minute, 40-second phone call?

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