Introduction A step function is a function that is a series of disconnected constant functions. This type of function is also referred to as a stair function because when graphed, the function resembles a staircase. Step functions are similar to piecewise functions in that these functions consist of several expressions over intervals combined into one function. A step function is useful when there are many cases for small continuous intervals of the domain. 6.2.2: Step Functions

Key Concepts A step function is a combination of one or more functions that are defined over restricted intervals, and which may be undefined at other domain points or over other restricted domain intervals. A common type of step function is one in which the function values are restricted to integer domain values, such as the number of students who join the computer-gaming club; each student must be represented by a whole number (integer) value. 6.2.2: Step Functions

Key Concepts, continued By contrast, a function that models temperature changes throughout the day would not be restricted to integer domain values, since temperatures can fluctuate by fractions of degrees. Two special types of step functions are called the floor function and the ceiling function. 6.2.2: Step Functions

Key Concepts, continued Floor Functions The floor function, sometimes known as the greatest integer function, can be written symbolically as . Floor functions are functions where for any input x, the output is the largest integer less than or equal to x. For example, is equal to 2 because 2 is the greatest integer that is less than or equal to 2.4. The following graph displays a floor function, 6.2.2: Step Functions

Key Concepts, continued Notice the domain (x-values) is all real numbers but the range (y-values) is restricted to integers. 6.2.2: Step Functions

Key Concepts, continued The floor function is not a continuous function. The open circles indicate where the function is not defined and the closed circles indicate where the function is defined. The function increases as the x-values approach positive infinity. Adding a constant to a floor function causes its graph to shift up or down depending on the sign of the function. For instance, for , adding c would shift the steps in the graph up by c units, while the steps in the graph of , would be shifted down by c units. 6.2.2: Step Functions

Key Concepts, continued Ceiling Functions The ceiling function, sometimes known as the least integer function, can be written symbolically as . Ceiling functions are where for any input x, the output is the smallest integer greater than or equal to x. For example, is equal to 4 because 4 is the least integer that is greater than or equal to 3.5. The following graph displays a ceiling function, 6.2.2: Step Functions

Key Concepts, continued Similar to a floor function, in this ceiling function, the domain (x-values) is all real numbers, but the range (y-values) is restricted to the set of all integers. 6.2.2: Step Functions

Key Concepts, continued The ceiling function is also not a continuous function. To graph a floor or ceiling function on a graphing calculator, follow the directions specific to your model. On a TI-83/84: Step 1: Press [Y=]. If anything is already entered, press [CLEAR]. Step 2: Press [MATH]. Arrow to “NUM,” then select 5: int(. Step 3: Enter the expression contained within the floor or ceiling function, using x as the variable. (continued) 6.2.2: Step Functions

Key Concepts, continued For a floor function, enter the remaining part of the expression by moving the cursor to the left and/or right of the int() expression. For example, “int(x+3)” is equivalent to the expression . For a ceiling function, enter a negative sign before the entire expression in the function brackets. Enter a negative sign before the int( function. Enter the remaining part of the expression by moving the cursor to the left and/or right of the int() expression. For example, “–int(–(x+3))” is equivalent to Step 4: Press [GRAPH]. 6.2.2: Step Functions

Key Concepts, continued On a TI-Nspire: Step 1: Press the [home] key. Step 2: Arrow over to the graphing icon and press [enter]. Step 3: Enter the expression at the bottom of the application, using x as the variable. Press the catalog key (to the right of the number 9). Select floor() to enter a greatest integer expression; for example, typing “floor(x+3)” is equivalent to the expression  . Select ceiling() to enter a least integer expression; for example, typing “ceiling(x+3)” is equivalent to the expression . Step 4: Press [enter] to plot the expression on the graph. 6.2.2: Step Functions

Common Errors/Misconceptions misinterpreting variables from a word problem  miscalculating with intervals in the step functions  incorrectly graphing the “steps” of step functions 6.2.2: Step Functions

Guided Practice Example 1 Consider the least integer (ceiling) function . How does the value of g(x) change as x changes from 1.5 to 3.8? Graph the function to show the change. 6.2.2: Step Functions

Guided Practice: Example 1, continued Evaluate g(x) at x = 1.5. A least integer function finds the smallest integer greater than or equal to the given value of x. So, when the function g(x) is evaluated at x = 1.5, the result is . 6.2.2: Step Functions

Guided Practice: Example 1, continued Evaluate g(x) at x = 3.8. The smallest integer greater than or equal to 3.8 is 4; therefore, . 6.2.2: Step Functions

Guided Practice: Example 1, continued Determine how the value of g(x) changes as x changes from 1.5 to 3.8. When the function g(x) is evaluated at x = 1.5, the result is 2; for x = 3.8, the result is 4. This is a difference of 4 – 2, or 2. The value of g(x) increases by 2 as x changes from 1.5 to 3.8. 6.2.2: Step Functions

Guided Practice: Example 1, continued Graph the function g(x) to show this change. The function can be graphed by hand or using a calculator. The result of either method will resemble the following graph. Notice that each expression is a horizontal line. Endpoints that are not included in the individual domain intervals are represented by an open circle. Endpoints that are included in individual domain intervals are represented by a closed circle. 6.2.2: Step Functions

Guided Practice: Example 1, continued Notice the increase of 2 units along the y-axis when x = 1.5 compared to when x = 3.8. This confirms the change in g(x) noted in step 3. ✔ 6.2.2: Step Functions

Guided Practice: Example 1, continued http://www.walch.com/ei/00745 6.2.2: Step Functions

Guided Practice Example 2 Consider the function . Graph f(x) and identify the domain and any points of discontinuity for this function. 6.2.2: Step Functions

Guided Practice: Example 2, continued Graph the function by hand or using a graphing calculator. The graph of is shown on the next slide. Notice that each expression is a horizontal line. Endpoints that are not included in the individual domain intervals are represented by open circles. Included endpoints in domain intervals are represented by closed circles. 6.2.2: Step Functions

Guided Practice: Example 2, continued 6.2.2: Step Functions

Guided Practice: Example 2, continued Identify the domain of f(x). Notice from the graph that the domain (set of all x-values) is all real numbers. 6.2.2: Step Functions

✔ Guided Practice: Example 2, continued Identify any points of discontinuity for f(x). The function is discontinuous for all integer values of x with points of discontinuity on the left end of each line segment when x = …, –2, –1, 0, 1, 2, 3, 4, 5, and so on. Some of the points of discontinuity are (0, 3), (1, 6), and (2, 9); however, if the graph’s axes were extended to show more values, there would be several more. Notice that the y-values increase by 3 each time, and 3 is the coefficient in the original equation . ✔ 6.2.2: Step Functions

Guided Practice: Example 2, continued http://www.walch.com/ei/00746 6.2.2: Step Functions