CEIL, FLOOR AND COMPOSITE FUNCTIONS

 CEIL , FLOOR AND COMPOSITE FUNCTION

CEIL AND FLOOR FUNCTION:-

INTRODUCTION:-

In Mathematics and Computer Programming, two important functions are used quite often. One is the floor function, and the other is the ceiling function For example, the floor and ceiling of a decimal 3.31 are 3 and 4 respectively.  So with the help of these two functions, we get the nearest integer in a number line of a given decimal.

CEILING FUNCTION:-

The ceiling function, denoted as $\lceil x\rceil ,\; is\; a\; mathematical\; function\; that\; rounds\; a\; real\; number$$x$ up to the smallest integer greater than or equal to $x$. In simpler terms, it gives the "smallest" integer that is not less than $x$.

Formal Definition:

For any real number $x$, the ceiling function $\lceil x \rceil$ is defined as:

$$\lceil x \rceil = \text{the smallest integer } n \text{ such that } n \geq x$$

Ceiling Function Symbol:

The ceiling function is also known as the smallest integer function. The notation to represent this function is ⌈ ⌉. It can be used as ⌈x⌉ or ceil (x) or f(x) = ⌈x⌉

The symbol of floor function is also a type of square bracket. i.e.,  ⌊ ⌋.

ON GRAPH:-

Ceiling Function Properties:

Let us consider that x and y are two real numbers and ceil (x) = ⌈x⌉. Some of the important properties of the ceiling functions are:

• ⌈x⌉ + ⌈y⌉ – 1 ≤ ⌈x + y⌉ ≤ ⌈x⌉ + ⌈y⌉
• ⌈x + a⌉ = ⌈x⌉ + a
• ⌈x⌉ = a; iff x ≤ a < x + 1
• ⌈x⌉ = a; iff x – 1 < a ≤ x
• a < ⌈x⌉ iff a < x
• a ≤ ⌈x⌉ iff x < a

Applications of the Ceiling Function:

The ceiling function is used extensively in various areas of mathematics, computer science, and real-life applications:

1. Computer Science:

   • In computer algorithms, the ceiling function is often used when you need to allocate enough resources to handle cases where division or other operations yield fractional results. For example, if you want to divide $15$ items among $4$ people, you'd need $\lceil 15 / 4 \rceil = 4$ groups to ensure everyone gets an equal share, even if it results in some leftover.

2. Geometry:

   • The ceiling function helps in calculating the number of tiles or bricks needed to cover an area. For example, if the area is $\mathrm{5.5\; square\; units\; and\; each\; tile\; covers }$$1$ square unit, you’d need $\lceil 5.5 \rceil = 6$tiles to cover the whole area.

3. Financial Calculations:

   • In financial transactions, such as interest calculations or payment installments, the ceiling function is often used to ensure that rounding results in enough resources to cover expenses.

4. Scheduling:

   • When planning schedules, the ceiling function is used to ensure that the minimum number of complete units of time are allocated. For example, if an event requires $3.2$ hours, the ceiling function would allocate$\lceil 3.2\rceil =\mathrm{4\; hours\; to\; ensure\; the\; event\; fits\; within\; whole\; hour\; slots.}$

5. Mathematics:

• In combinatorics, number theory, and probability, the ceiling function is used in calculations involving bounds, ensuring that values are rounded in a way that satisfies problem constraints.

Real-Life Applications of the Ceiling Function:

1. Elevator Capacity:

   • Scenario: You are designing an elevator, and you need to know how many trips it will take to move 53 people if the elevator can carry a maximum of 10 people per trip.
   • Solution: Using the ceiling function, $\lceil 53\mathrm{/}10\rceil =\mathrm{6,\; you\; conclude\; that\; the\; elevator\; will\; need\; to\; make\; 6\; trips\; to\; carry\; all\; the\; passengers,\; rounding\; up\; because\; the\; elevator\; cannot\; make\; half\; a\; trip.}$

2. Packaging Products:

   • Scenario: You are packing products in boxes, and each box can hold up to 8 items. You need to find out how many boxes you need if you have 67 items.
   • Solution: $\lceil 67/8 \rceil = 9$, so you will need 9 boxes in total because one box will need to hold the remaining items.

3. Seating Arrangements:

   • Scenario: You are organizing a conference and want to seat guests in rows, with each row holding 15 seats. If you have 132 guests, how many rows do you need?
   • Solution: By applying the ceiling function, $\lceil 132\mathrm{/}15\rceil =\mathrm{9,\; you\; determine\; that\; you\; need\; 9\; rows\; to\; seat\; everyone,\; ensuring\; that\; no\; one\; is\; left\; standing.}$

4. Budgeting and Financial Planning:

   • Scenario: You are calculating the budget for a project, and certain costs, such as equipment rentals, are charged on a daily basis. If your project will last 2.5 days, you need to pay for 3 full days.
   • Solution: Using the ceiling function, $\lceil 2.5 \rceil = 3$, you budget for 3 days of rental fees, as partial days are billed as full days.

5. Downloading or Uploading Files:

   • Scenario: You are uploading files, and each file takes 2.3 MB of storage. You want to calculate how many 5 MB storage units you will need to store a 100 MB file.
   • Solution: Using the ceiling function $\lceil 100\mathrm{/}5\rceil =\mathrm{20,\; you\; know\; you\; will\; need\; at\; least\; 20\; storage\; units,\; since\; a\; fraction\; of\; a\; unit\; cannot\; be\; used.}$

Conclusion:

The ceiling function is a fundamental mathematical concept that ensures rounding up to the nearest integer. It has wide applications in both theoretical and practical domains, making it an essential tool for rounding operations where approximations toward higher values are required.

FLOOR FUNCTION:

The floor function, denoted as $\lfloor x \rfloor$, is a mathematical function that rounds a real number $x$ down to the nearest integer less than or equal to $x$. Unlike the ceiling function, which rounds numbers up, the floor function rounds down to the largest integer smaller than or equal to $x$.

Formal Definition:

For any real number $x$, the floor function $\lfloor x \rfloor$ is defined as:

$$\lfloor x\rfloor =\text{the greatest integer }n\text{ such that }n\le x$$

In other words, $\lfloor x \rfloor$ gives the largest integer that is less than or equal to $x$.

Examples:

To understand how the floor function works, here are some examples:

• $\lfloor 4.7\rfloor =4$
  • The largest integer less than or equal to $4.7$ is $4$.
• $\lfloor -2.3 \rfloor = -3$
  
  • The largest integer less than or equal to $-2.3$ is $-3$.
• $\lfloor 5\rfloor =5$
• If $x$ is already an integer, the floor function simply returns $\mathrm{x.}$

Graphical Representation:

Graphically, the floor function is a step function. The function remains constant over intervals of the form $[n, n+1)$, where $n$ is an integer. The function jumps downward at each integer value.

For example:

• For $0 \leq x < 1$, $\lfloor x \rfloor = 0$.
• For $1 \leq x < 2$, $\lfloor x \rfloor = 1$, and so on.

On line graph:-

On graph:-

Floor Function Properties:

The important properties of greatest integer function are:

• ⌊x⌋ = x, where x is an integer
• ⌊x + n⌋ = ⌊x⌋ + n, where n ∈ Z
• ⌊-x] = –⌊x], if x ∈ Z
• ⌊-x] =-⌊x] – 1, if x ∉ Z
• If ⌊f(x)] ≥ Y, then f(x) ≥ Y

Note: Z stands for set of integers

Applications of the Floor Function:

The floor function has many practical applications in both mathematics and real-world scenarios:

1. Computer Science:

   • The floor function is widely used in algorithms, especially when dividing tasks, calculating indexes in arrays, or determining resource allocation. For example, when you divide $15$ items among $4$ people, $\lfloor 15 / 4 \rfloor = 3$ indicates that each person gets at least $3$ items.

2. Number Theory:

   • In number theory, the floor function is used in a variety of formulas and theorems, such as in the analysis of sums involving fractions. It is used to define the greatest integer function and has applications in counting problems and modular arithmetic.

3. Mathematics and Geometry:

   • In geometry, the floor function is used when calculating the number of units required to cover a space. For example, if you want to cover a length of $7.8$ meters with tiles that are $1$ meter long, you will need $\lfloor 7.8 \rfloor = 7$ whole tiles.

4. Financial Calculations:

   • In financial applications, the floor function is often used to determine the lower bound of costs or payments. For example, if a payment of $\$10.75$ needs to be processed in whole dollars, the floor function would round it down to $\$10$.

5. Scheduling:

   • In scheduling, the floor function helps in determining how many full time intervals are required for an event. For instance, if an event takes $2.5$ hours, you might allocate $\lfloor 2.5 \rfloor = 2$ full hours for it.

6. Limits and Series:

• In calculus and analysis, the floor function can appear in the study of limits, series, and sums. It helps in partitioning domains or setting bounds for sums and integrals.

Real-Life Applications of the Floor Function:

1. Resource Allocation:

   • Scenario: You have 47 people to assign to 4 buses, and you want to figure out how many people can fit in each bus if you distribute them as evenly as possible.
   • Solution: By using the floor function, $\lfloor 47\mathrm{/}4\rfloor =\mathrm{11,\; you\; know\; that\; each\; bus\; can\; accommodate\; at\; least\; 11\; people,\; with\; the\; remaining\; 3\; people\; distributed\; among\; a\; few\; buses.}$

2. Cost Rounding Down:

   • Scenario: In e-commerce or financial transactions, you may need to calculate the amount a customer should be billed after applying a discount, but you can only charge in whole numbers.
   • Solution: If the calculated amount is $10.75, using the floor function $\lfloor 10.75 \rfloor = 10$, the system rounds down to charge $10.

3. Time Scheduling:

   • Scenario: You want to schedule meetings and need to determine how many full hours are available in a given timeframe.
   • Solution: If a meeting lasts 1.5 hours and you have 7 hours available, $\lfloor 7/1.5 \rfloor = 4$, meaning you can schedule 4 full meetings in that period.

4. Tiling a Floor:

   • Scenario: You need to tile a rectangular floor that measures 9.2 meters by 4.7 meters using square tiles that are 1 meter long.
   • Solution: To calculate how many tiles are needed along each dimension, you use the floor function for the length $\lfloor 9.2\rfloor =\mathrm{9\; and\; for\; the\; width }$$\lfloor 4.7\rfloor =\mathrm{4.\; This\; tells\; you\; that\; you\; can\; fit\; 9\; full\; tiles\; in\; one\; direction\; and\; 4\; in\; the\; other,\; covering\; the\; space.}$

Conclusion:

The floor function is a fundamental mathematical operation that provides the greatest integer less than or equal to a given real number. It has numerous applications in mathematics, computer science, geometry, and practical problem-solving scenarios where rounding down is required. The floor function plays a crucial role in various fields by providing an easy way to handle non-integer values and ensuring results align with practical constraints

COMPOSITE FUNCTIONS:

A composite function is a function that is created by applying one function to the result of another function. It involves the composition of two functions to form a new function, where the output of one function becomes the input of another. The composition of functions is an important concept in mathematics, particularly in algebra and calculus.

Definition of Composite Function:

If you have two functions:

• $f: A \to B$ (a function from set $\mathrm{A\; to\; set }$$B$)
• $g: B \to C$ (a function from set $\mathrm{B\; to\; set }$$\mathrm{C)}$

Then the composite function $g\circ \mathrm{f\; (read\; as\; "g\; of\; f")\; is\; defined\; as:}$

$$(g\circ f)(x)=g(f(x))$$

In this case:

• $f(x)$ is evaluated first, producing an output.
• Then, $g$ is applied to the output of $f(x),\; producing\; the\; final\; result.$
• The domain of $g \circ f$is the set of all $x \in A$ such that $f(x)is\; in\; the\; domain\; of$$\mathrm{g.}$

Notation:

• The composition of two functions $f$and $g$ is written as $g \circ f$, and this is read as "g composed with f" or "g of f."
• It is crucial to note that $g \circ f$ is not the same as $f\circ \mathrm{g.\; The\; order\; in\; which\; the\; functions\; are\; composed\; matters.}$

Understanding Composite Function:

In a composite function $g\circ \mathrm{f,\; we\; start\; by\; applying\; the\; first\; function }$$f$, and then we apply the second function $g$ to the result. In simpler terms:

$$g(f(x)) = \text{"apply function } f \text{ to } x \text{, then apply function } g \text{ to the result.}$$

Example of Composite Function:

Let’s consider two functions:

• $f(x) = x^2$ (a function that squares its input)
• $g(x) = 2x + 1$ (a function that multiplies its input by 2 and adds 1)

Now, let's compute $g\circ \mathrm{f:}$

$$(g \circ f)(x) = g(f(x)) = g(x^2) = 2(x^2) + 1 = 2x^2 + 1$$

Here’s what happened step by step:

1. First, $f(x) = x^2$ is applied, which squares $x$.
2. Then, $g(x^2)=2x^2+\mathrm{1\; is\; applied\; to\; the\; result\; of }$$f(x)$, giving the final composite function $g \circ f(x) = 2x^2 + 1$.

If we reverse the order and compute $f\circ \mathrm{g:}$

$$(f \circ g)(x) = f(g(x)) = f(2x + 1) = (2x + 1)^2 = 4x^2 + 4x + 1$$

So, $f \circ g(x) = 4x^2 + 4x + 1$ is not the same as $g \circ f(x) = 2x^2 + 1$. This shows that the order of composition matters.

Graphical Interpretation of Composite Functions:

Graphically, composite functions can be understood by applying one transformation after another. Imagine two functions that modify a graph:

1. $f(x)$ applies the first transformation to $x$.
2. $g(f(x))\; then\; applies\; the\; second\; transformation\; to\; the\; result\; of$$f(x)$.

For example, if $f(x)$ represents a squaring function and $g(x)$ represents a linear transformation, the graph of $g(f(x))$ will show a parabola that has been stretched and shifted by the linear transformation $g$.

Applications of Composite Functions:

Composite functions are widely used in various areas of mathematics, science, and engineering:

1. Mathematical Modeling:

   • In many real-world problems, a situation can be described by a series of steps or processes. Each step can be represented by a function, and the entire process can be described by composing these functions. For example, in economics, the demand function and price function can be combined to form a composite function that represents revenue.

2. Chain Rule in Calculus:

   • In calculus, the chain rule for derivatives involves composite functions. If $y = g(f(x))$, then the derivative of $y$ with respect to $x$ is given by: $$\frac{dy}{dx} = \frac{dg}{df} \cdot \frac{df}{dx}$$ This rule allows us to differentiate composite functions efficiently.

3. Control Systems:

   • In control systems, different processes (like feedback loops) are modeled using functions. Composing these functions helps in understanding the overall behavior of the system.

4. Physics and Engineering:

   • Composite functions are used in physics to model various phenomena. For example, in kinematics, the position of a particle as a function of time might depend on the velocity, which in turn depends on time. Thus, the position function can be expressed as a composite of time-dependent functions.

5. Computer Science:

   • In programming and algorithms, composite functions are often used to combine multiple operations. For instance, in data processing pipelines, a sequence of operations is applied to data, and each operation can be seen as a function, with the entire pipeline being a composition of these functions.

Example in Real Life:

Suppose you are converting temperatures from Celsius to Fahrenheit and then to Kelvin. These conversions can be represented as functions:

• $f(C) = \frac{9}{5}C + 32$(Celsius to Fahrenheit)
• $g(F) = \frac{5}{9}(F - 32) + 273.15$ (Fahrenheit to Kelvin)

To find a composite function that converts directly from Celsius to Kelvin, you would compute $g \circ f(C)$, which gives you the complete transformation without intermediate steps.

Conclusion:

Composite functions are a powerful tool for simplifying complex processes by combining simpler functions. They play a crucial role in many mathematical areas, including calculus, algebra, and applied fields such as engineering and computer science. Understanding how to work with and manipulate composite functions is essential for solving a wide variety of problems involving multi-step processes.

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NAME- SWARA ARYA

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COURSE- 1BDA(BSC in data science and artificial intelligence)

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