Step Function Modeling

Constant Outputs on Separate Intervals

A step function has a constant value on each interval and may jump when one interval ends and the next begins. Its graph resembles a staircase: horizontal segments represent the constant values, while separate endpoints mark any jump discontinuities.

Definition of a Step Function

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

In this definition, $c_1, c_2, c_3, \ldots, c_n$ 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

Step functions stay constant on each interval and jump at the boundaries. The floor and ceiling functions differ in which side of the boundary the new value appears, and the Heaviside step records a single jump at zero.

Floor Function

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

The graph marks each included left endpoint with a filled point and each excluded right endpoint with an open point.

Floor function value table:

Ceiling Function

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

The graph marks each excluded left endpoint with an open point and each included right endpoint with a filled point.

Heaviside Unit Step Function

The examples below use a common convention for the unit step function or Heaviside function, denoted by $H(x)$ or $u(x)$:

Under this convention, the unit step function has value $0$ for $x < 0$ and $1$ for $x \geq 0$. Other fields sometimes choose $H(0)=0$ or $H(0)=\frac12$, so the value at the jump must always be stated.

Properties of Step Functions

General properties:

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

Special properties of floor and ceiling functions:

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

Comparison table of floor and ceiling functions:

Transformations of Step Functions

A shift moves every jump by the same amount, while a scaling changes the distances between the jumps.

Shifting or scaling a step function moves its jumps.

Vertical Translation

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

Horizontal Translation

The function $f(x) = \lfloor x - h \rfloor$ shifts the graph to the right (if $h > 0$) or to the left (if $h < 0$). For example, $g(x) = \lfloor x - 2 \rfloor$ moves every step of the floor function two units to the right, so the first jump lands at $x = 2$ instead of $x = 0$. Replacing $h$ with a negative value moves the jumps the other way.

Practice Problems

The problems mix floor and ceiling values with a quantity that changes in whole steps. Evaluate the whole-number part before combining the terms.

1. Determine the value of $\lfloor 3.7 \rfloor + \lceil -2.3 \rceil$

2. A bookstore gives discounts based on the number of purchases:

• $1\text{-}5$ books: no discount

• $6\text{-}10$ books: $10\%$ discount

• $11\text{-}20$ books: $15\%$ discount

• $> 20$ books: $20\%$ discount

  If the price per book is $\text{Rp}50{,}000$, create a function that represents the total price after discount!

3. Graph of function $f(x) = 2\lfloor x \rfloor - 1$ for $-3 \leq x \leq 3$

4. Solve the equation $\lfloor 2x + 1 \rfloor = 5$

5. An elevator can accommodate a maximum of $8 \text{ people}$. If there are $n \text{ people}$ who want to use the elevator, how many times must the elevator operate?

Answer Key

1. Calculating floor and ceiling function values:

   $$\lfloor3.7\rfloor=3$$

   $3$ is the largest integer less than or equal to $3.7$.

   $$\lceil-2.3\rceil=-2$$

   $-2$ is the smallest integer greater than or equal to $-2.3$.

   $$\lfloor 3.7 \rfloor + \lceil -2.3 \rceil = 3 + (-2) = 1$$

2. Bookstore discount function model:

   Let $n\in\mathbb{N}$ be the positive whole number of books purchased. The total price after discount is:

   $$H(n) = \begin{cases}\begin{gathered}50000n\\\text{if } 1 \leq n \leq 5\end{gathered}\\[4pt]\begin{gathered}50000n \times 0.9\\\text{if } 6 \leq n \leq 10\end{gathered}\\[4pt]\begin{gathered}50000n \times 0.85\\\text{if } 11 \leq n \leq 20\end{gathered}\\[4pt]\begin{gathered}50000n \times 0.8\\\text{if } n > 20\end{gathered}\end{cases}$$

3. Graph of function $f(x) = 2\lfloor x \rfloor - 1$:

   For each interval:

   • $-3 \leq x < -2: f(x) = 2(-3) - 1 = -7$
   • $-2 \leq x < -1: f(x) = 2(-2) - 1 = -5$
   • $-1 \leq x < 0: f(x) = 2(-1) - 1 = -3$
   • $0 \leq x < 1: f(x) = 2(0) - 1 = -1$
   • $1 \leq x < 2: f(x) = 2(1) - 1 = 1$
   • $2 \leq x < 3: f(x) = 2(2) - 1 = 3$
   • At the endpoint, $f(3)=2\lfloor3\rfloor-1=5$

   The following graph shows the function value on each interval and at the endpoints:

   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 $-1$.

   GridAutomatic rotation

4. 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 $x \in [2, 2.5)$.

5. Calculating the number of elevator operations:

   If $n\in\mathbb{Z}_{\geq0}$ is the number of people and the elevator can accommodate at most $8 \text{ people}$, then the number of elevator trips required is:

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

   The ceiling function is used because any nonzero remainder requires one additional trip. It also gives $f(0)=0$ when nobody is waiting.