Step Function Modeling

Understanding Step Functions

A step function is a type of mathematical function that has constant values on certain intervals and experiences sudden jumps at specific points. The graph of this function resembles stairs, with horizontal lines connecting points of discontinuity.

Mathematical Definition

A step function can be defined as a piecewise function of the form:

f(x) = \begin{cases} c_1, & \text{if } a_1 \leq x < b_1 \\ c_2, & \text{if } a_2 \leq x < b_2 \\ c_3, & \text{if } a_3 \leq x < b_3 \\ \vdots \\ c_n, & \text{if } a_n \leq x < b_n \end{cases}

where $c_1, c_2, c_3, \ldots, c_n$ are constants and $[a_i, b_i)$ are non-overlapping intervals.

Characteristics of step functions:

Constant values on each interval
Jump discontinuities at interval boundary points
Graph shaped like stairs
Belongs to the category of piecewise functions

Types of Step Functions

Floor Function

The floor function, denoted by $\lfloor x \rfloor$ , gives the largest integer less than or equal to $x$ .

\lfloor x \rfloor = \max\{n \in \mathbb{Z} : n \leq x\}

To make it easier to understand, let's look at the following example:

Floor Function

f(x) = \lfloor x \rfloor

The graph shows the floor function that gives the largest integer value that does not exceed x.

Floor function value table:

x	-2.5	-1.7	-1	-0.3	0	0.8	1	1.9	2.4
$\lfloor x \rfloor$	-3	-2	-1	-1	0	0	1	1	2

Ceiling Function

The ceiling function, denoted by $\lceil x \rceil$ , gives the smallest integer greater than or equal to $x$ .

\lceil x \rceil = \min\{n \in \mathbb{Z} : n \geq x\}

Let's look at the following example:

Ceiling Function

f(x) = \lceil x \rceil

The graph shows the ceiling function that gives the smallest integer value that is not less than x.

Unit Step Function (Heaviside)

The unit step function or Heaviside function, denoted by $H(x)$ or $u(x)$ , is defined as:

H(x) = \begin{cases} 0, & \text{if } x < 0 \\ 1, & \text{if } x \geq 0 \end{cases}

Unlike the floor and ceiling functions, the unit step function has a value of 0 for $x < 0$ and 1 for $x \geq 0$ .

Unit Step Function

H(x)

The graph shows the unit step function that jumps from 0 to 1 at point x = 0.

Properties of Step Functions

General properties:

Domain: $\mathbb{R}$ (usually)
Range: Set of discrete values
Continuity: jump discontinuities at certain points

Special properties of floor and ceiling functions:

$\lfloor x \rfloor \leq x < \lfloor x \rfloor + 1$
$\lceil x \rceil - 1 < x \leq \lceil x \rceil$
$\lceil x \rceil = \lfloor x \rfloor + 1 \text{ if } x \notin \mathbb{Z}$
$\lceil x \rceil = \lfloor x \rfloor \text{ if } x \in \mathbb{Z}$

Comparison table of floor and ceiling functions:

x	$\lfloor x \rfloor$	$\lceil x \rceil$	Difference
-2.3	-3	-2	1
-1	-1	-1	0
0.7	0	1	1
2	2	2	0
3.8	3	4	1

Transformations of Step Functions

Vertical Translation

The function $f(x) = \lfloor x \rfloor + k$ shifts the floor function graph upward (if $k > 0$ ) or downward (if $k < 0$ ).

Vertical Translation of Floor Function

Comparison of

f(x) = \lfloor x \rfloor

with

g(x) = \lfloor x \rfloor + 2

and

h(x) = \lfloor x \rfloor - 1

Horizontal Translation

The function $f(x) = \lfloor x - h \rfloor$ shifts the graph to the right (if $h > 0$ ) or to the left (if $h < 0$ ).

Practice Problems

Determine the value of $\lfloor 3.7 \rfloor + \lceil -2.3 \rceil$
A bookstore gives discounts based on the number of purchases:
- 1-5 books: no discount
- 6-10 books: 10% discount
- 11-20 books: 15% discount
- 20 books: 20% discount
If the price per book is Rp 50,000, create a function that represents the total price after discount!
Graph of function $f(x) = 2\lfloor x \rfloor - 1$ for $-3 \leq x \leq 3$
Solve the equation $\lfloor 2x + 1 \rfloor = 5$
An elevator can accommodate a maximum of 8 people. If there are $n$ people who want to use the elevator, how many times must the elevator operate?

Answer Key

Calculating floor and ceiling function values:

$\lfloor 3.7 \rfloor = 3 \text{ (largest integer } \leq 3.7\text{)}$
$\lceil -2.3 \rceil = -2 \text{ (smallest integer } \geq -2.3\text{)}$
$\lfloor 3.7 \rfloor + \lceil -2.3 \rceil = 3 + (-2) = 1$
Bookstore discount function model:

Let $n$ be the number of books purchased, then the total price after discount is:

$H(n) = \begin{cases} 50000n, & \text{if } 1 \leq n \leq 5 \\ 50000n \times 0.9, & \text{if } 6 \leq n \leq 10 \\ 50000n \times 0.85, & \text{if } 11 \leq n \leq 20 \\ 50000n \times 0.8, & \text{if } n > 20 \end{cases}$
Graph of function $f(x) = 2\lfloor x \rfloor - 1$ :

For each interval:
- $-3 \leq x < -2: f(x) = 2(-3) - 1 = -7$
- $-2 \leq x < -1: f(x) = 2(-2) - 1 = -5$
- $-1 \leq x < 0: f(x) = 2(-1) - 1 = -3$
- $0 \leq x < 1: f(x) = 2(0) - 1 = -1$
- $1 \leq x < 2: f(x) = 2(1) - 1 = 1$
- $2 \leq x \leq 3: f(x) = 2(2) - 1 = 3$
If we create the graph, it would look approximately like the following:

Graph of $f(x) = 2\lfloor x \rfloor - 1$
The graph shows the transformation of the floor function with a scale factor of 2 and vertical translation of -1.
Solving the equation $\lfloor 2x + 1 \rfloor = 5$ :

$5 \leq 2x + 1 < 6$
$4 \leq 2x < 5$
$2 \leq x < 2.5$

So the solution set is $x \in [2, 2.5)$ .
Calculating the number of elevator operations:

If there are $n$ people and the elevator can accommodate a maximum of 8 people, then the number of elevator operations required is:

$f(n) = \lceil \frac{n}{8} \rceil$

The ceiling function is used because if there are remaining people (less than 8), one additional elevator operation is still required.

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