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.

IntroductionA 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.

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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.

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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.

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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,

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Key Concepts, continued

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• Notice the domain (x-values) is all real numbers but the range (y-values) is restricted to integers.

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


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,

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Key Concepts, continued

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• 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.

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.

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(continued)

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].

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Key Concepts, continuedOn 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.11


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

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Guided Practice

Example 1Consider 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.

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Guided Practice: Example 1, continued

1. 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 .

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2. Evaluate g(x) at x = 3.8.

The smallest integer greater than or equal to 3.8 is 4;

therefore, .

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3. 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.

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4. 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.

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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 instep 3.


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Guided Practice

Example 2Consider the function . Graph f(x) and identify the domain and any points of discontinuity for this function.

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1. 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.

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2. Identify the domain of f(x). Notice from the graph that the domain (set of all x-values) is all real numbers.

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3. 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 theoriginal equation .

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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.

Documents

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continuous function

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stair function

step functions slide

greatest integer function

common type of step