Mathematics 🔢

Pin Your Way to A+ Grades.

01 Activity Questions 1.1

1. Below is a list of numbers: 22, -17, 47, -2000, 0, 1, 43, 1729, 6174, -63, 100, 32, -9. How many natural numbers are there in the given list? a) 6 b) 7 c) 8 d) 9 Solution Based on the sources, the set of natural numbers is denoted by N. This set includes 0, 1, 2, 3, 4, 5, and so on. The sources explicitly state that whenever they are talking about natural numbers, it always includes a 0, even though some books may not. Natural numbers are primarily used for counting.

🧠

02 natural numbers and the operations

Here’s an explanation of natural numbers and their operations in a notes format, designed for ease of understanding with emojis: Natural Numbers and Operations 🔢 Natural Numbers (N) Natural numbers are primarily used for counting objects (e.g., 7 balls ⚽️⚽️⚽️⚽️⚽️⚽️⚽️, 7 pencils ✏️✏️✏️✏️✏️✏️✏️). They are denoted by the symbol N. In this context, the set of natural numbers includes 0 (i.e., {0, 1, 2, 3, 4, …}). 💡 Historically, there can be some confusion in different books about whether 0 is included, but here, it always is. Arithmetic Operations

🧠

03 Set theory

Here’s an explanation of set theory, presented in a notes format with emojis for easy understanding: Set Theory 📚 What is a Set? At its most basic level, a set is a collection of items. These items are called elements or members. Sets can be finite (e.g., days of the week 🗓️, months in a year) or infinite (e.g., the set of integers 🔢). Key Characteristics of Sets ✨

🧠

04 Activity Questions 1.2

1. Which of the following option(s) is(are) true? Solution Based on the sources and our conversation history, we can determine which of the given inequalities between fractions are true by finding a common denominator and comparing the numerators. Rational numbers, which include fractions, can be written in the form p/q. To compare two fractions which have different denominators, there is no way to directly compare them. The only way is to convert them into equivalent fractions such that they have the same denominator. A number that is a multiple of both denominators can be used as the common denominator. Once the denominators are the same, you can add the numerators, or in this case, compare them.

🧠

05 Rational numbers

Here’s an explanation of rational numbers, presented in a notes format with emojis for easy understanding: Rational Numbers (Q) 💭 What are they? Rational numbers are numbers that can be written as a ratio of two integers. They are expressed in the form p/q ➗, where ‘p’ and ‘q’ are both integers. The top number, ‘p’, is called the numerator ⬆️, and the bottom number, ‘q’, is called the denominator ⬇️. The set of rational numbers is denoted by the symbol Q. 💡 The ‘Q’ stands for quotient, linking to the idea of a ratio. Extending from other number sets

🧠

06 Activity Questions 1.3

1. Which of the following statement(s) is(are) false? a) The sum of two natural numbers is always a natural number b) The difference between two integers is always an integer c) The product of two rational numbers is always a real number d) The product of two irrational numbers is always an irrational number Solution The statement that is false is:

🧠

07 real and complex numbers

Here’s an explanation of real numbers and complex numbers in a notes format, designed for ease of understanding with emojis: Real Numbers (R) 🌍 Real numbers are an expansion of rational numbers and fill up the entire number line 📏, including all the “gaps” that rational numbers leave. They are denoted by the symbol R. What fills the gaps? Irrational Numbers 💫 Irrational numbers are those that cannot be written as a simple fraction p/q, where p and q are integers. They are simply numbers that are not rational. A classic example is the square root of 2 (√2). You can physically draw a line segment of length √2 (e.g., the hypotenuse of a square with sides of length 1). However, it cannot be precisely expressed as a ratio of two integers. This fact was known to ancient Greeks like Pythagoras, and its irrationality was reportedly proved by his follower Hippasus around 500 BCE, shocking the Pythagoreans who believed rational numbers formed the basis of all science. In general, the square root of any integer that is not a perfect square (e.g., √3, √5, √6) is an irrational number. Other well-known irrational numbers include pi (π) (the ratio of a circle’s circumference to its diameter) and e (used in natural logarithms). These numbers have infinite non-repeating decimal expansions. Density Property 🌊 Just like rational numbers, real numbers are dense: you can always find another real number between any two distinct real numbers (for example, by taking their average). This means there are no “gaps” in the real number line. Relationship to other Number Sets 🌳 Every natural number is an integer, every integer is a rational number, and every rational number is a real number. The set of natural numbers (N) is a subset of integers (Z). The set of integers (Z) is a subset of rational numbers (Q). The set of rational numbers (Q) is a proper subset of real numbers (R). This means that while all rational numbers are real numbers, there are real numbers (the irrationals) that are not rational. This hierarchical relationship can be visualized using Venn diagrams, where N is the innermost circle, followed by Z, then Q, and finally R as the largest encompassing circle. “Size” of Infinity ✨ Even though rational numbers are dense, the set of real numbers has a larger “size” or cardinality of infinity than the set of natural numbers, integers, or rational numbers. This implies there are vastly more irrational numbers than rational numbers. Complex Numbers (C) 🌌 The Need for Expansion 🚧 When dealing with operations like square roots of negative numbers, the existing real number system falls short. For instance, if you try to find the square root of -1 (√-1), you cannot find a real number that, when multiplied by itself, yields a negative result. This is because the rule for multiplication of signs states that if two numbers have the same sign (either both positive or both negative), their product is always positive. This limitation is also seen when solving quadratic equations: if the discriminant (b² - 4ac) is less than 0, it means you’d be taking the square root of a negative number, which implies no real solutions. Introducing Complex Numbers ✨ To allow for the square roots of negative numbers, a new class of numbers called complex numbers was created. Complex numbers extend the real number system. Symbol ℂ While the provided sources do not explicitly state a symbol for complex numbers, they follow a pattern of using single letters (N, Z, Q, R) for other number sets. Complex numbers are commonly denoted by C (or ℂ). This information is not directly from the provided sources and you may want to independently verify it. Beyond this Course 📚 The provided sources indicate that the study of complex numbers is generally beyond the scope of this particular course.

🧠

08 Activity Questions 1.4

Q1. Which of the following sets are same? (i) {Ankitha, Keerthana, Raju, Suresh} (ii) {Raju, Ankitha, Keerthana, Raju, Ankitha, Suresh} (iii) {Keerthana, Suresh, Dheeraj, Raju, Ankitha} (iv) {Suresh, Raju, Ankitha, Keerthana} (v) {Dheeraj, Raju, Soumya, Keerthana} a) (i) and (ii) b) (iii),(iv) and (v) c) (i) and (iv) d) (i),(ii) and (iv) Solution Q2. Suppose X = {3, π, Tiger, Ball, -40, Dhoni}. Which of the following statement(s) is(are) true about X?

🧠

09 Activity Questions 1.5

Question 1 Which of the following is a correct representation of set comprehension? ○ {x ; x ∈ ℕ, x is even} ○ {x | x ∈ ℕ, x is even} ○ {x is even | x ∈ ℕ | x} ○ {x is even ; x ∈ ℕ, x} Solution Question 2 Which of the following is the set of natural numbers that are multiples of 3 or 5?

🧠

10 construction of subsets and set operation

Here’s an explanation of how subsets are constructed and common set operations, presented in a notes format for easy understanding with emojis: Construction of Subsets (Set Comprehension) 🛠️ What it is Set comprehension is a notation used to define subsets from existing sets. It’s especially useful for infinite sets where you can’t just list all the elements explicitly. It provides a precise and compact way to describe a set. Think of it like a recipe 🧑‍🍳 for building a new set by selecting items from an existing one based on certain rules. The Three Main Parts ✨ Set comprehension generally involves three components:

🧠

11 Activity Questions 1.6

Q1. Which of the following sets is(are) infinite? Set of all Indian Nobel laureates Set of squares of all odd natural numbers Set of all countries in the world Set of all leap years Solution Q2. Which of the following set comprehension defines real numbers in interval [2, 0) ∪ (4, 8]?

🧠

12 Sets - Examples

Hello! I can certainly help you understand sets better. Here’s an explanation in a notes format, complete with emojis and practice questions, based on the provided sources: 📝 Understanding Sets: Your Quick Guide! What is a Set? 🤝 A set is simply a collection of well-defined items. Think of it like a neatly organised group of things! 📦 For example, the days of the week are a set: {Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday} 🗓️. Elements of a Set 🌟 The items within a set are called elements or members.

🧠

13 Activity Questions 1.7

Q1. In a class of 40 students, 20 took Mathematics, 18 took Statistics, 16 took English, 7 took Mathematics and Statistics, 7 took Mathematics and English, 9 took Statistics and English, and 3 took all the three courses. How many students are not taking any of these courses? 3 4 5 6 Solution Q2. In a survey, it is found that 30 people like apples, 45 people like oranges, and 20 people like both apples and oranges. How many people like only one fruit? Solution Q3. In a class, 50 students play cricket, 35 play football, 14 play both, and 9 play neither. How many students are there in the class?

🧠

14 natural numbers and the operations

Based on the sources and our conversation history, a set is fundamentally understood as a collection of items. These items are called members or elements of the set. A crucial aspect of sets is that the order in which the members are listed does not matter, and duplicate members do not change the set. The cardinality of a set is the number of items or elements it contains. For finite sets, this is a straightforward count.

🧠

15 Activity Questions 1.8

Q1. Let A = {1, 4} and B = {2, 4, 6, 8}. Which of the following is the Cartesian product of A and B? ○ {(1, 2), (1, 4), (1, 6), (1, 8), (4, 1), (4, 4), (4, 6), (4, 8)} ○ {(1, 2), (1, 4), (1, 6), (1, 8), (4, 2), (4, 4), (4, 6), (4, 8)} ○ {(1, 2), (1, 4), (1, 6), (1, 8), (2, 4), (4, 2), (4, 6), (4, 8)}

🧠

16 Relations

Hello there! Let’s dive into the fascinating world of relations, building upon our previous understanding of sets. 📝 Understanding Relations: Your Connection Guide! What is a Relation? 🤝 A relation is fundamentally a collection of ordered pairs. Think of it as a specific way to connect elements from two (or more) sets. More formally, a relation is a subset of a Cartesian product.

🧠

17 Activity Questions 1.9

Q1. Suppose f : ℤ → ℤ is a function defined by f(k) = k³ + 4k - 10. The value of f(k) at k = 4 is ______ This is a fill-in-the-blank question asking for a numerical answer. Solution Q2. Let f(x) = |x| + 5 and Dom(f) = {c ∈ ℝ | f(x) ∈ ℝ}. Which of the following is(are) true?

🧠

18 Functions

Here’s an explanation of functions, designed for ease of understanding with emojis, followed by practice questions and their answers in a notes format. 📝 Understanding Functions: Your Input-Output Machine! What is a Function? ⚙️ A function is a special kind of relation. While a relation is simply a collection of ordered pairs, a function is a rule that takes an input from one set and maps it to exactly one output in another set.

🧠

19 Activity Questions 1.10

Answer the questions 1-3, based on following information: Let A = {x|x ∈ ℕ, x < 10 and x is odd} B = {y|y ∈ ℕ, y is a perfect square and 15 < y < 40} Q1. Which of the following is a subset of B × A? ○ {(36, 3), (25, 5), (36, 6)} ○ {(1, 25), (6, 36), (7, 25), (3, 36)} ○ {(16, 5), (25, 9), (36, 3), (16, 1)}

🧠

20 Relations - Examples

Here’s an explanation of relations, keeping ease of understanding and emojis in mind, formatted as notes, drawing directly from the sources provided. 📝 Understanding Relations: Flexible Connections! What is a Relation? 🤝 At its core, a relation is a collection of ordered pairs. It’s a way to describe how elements from one set connect to elements in another set (or even within the same set).

🧠

21 Activity Questions 1.11

Q1. If Dom(f) = {x ∈ ℝ, f(x) ∈ ℝ} defined by f(x) = (x + 12)/(4x - 8), then the domain of the function f is ______ ○ ℝ ○ ℝ \ {1/4} ○ ℝ \ {-12} ○ ℝ \ {2} Solution Q2. The product of the minimum value of the function f(x) = 9|x| - 8 and the maximum value of the function g(x) = 11 - |x + 8| is ______

🧠

22 FUnctions - Examples

Based on the sources and our conversation history, functions are described as a special type of relation where each element in the input set is mapped to exactly one element in the output set. Abstractly, a function can be thought of as a machine that produces an output for a given input. Here are several examples of functions found in the sources: Functions on Numbers: The Square Function: This is a frequently used example. Given an input x, it returns x². It can be written as f(x) = x². The domain and codomain are often considered the set of real numbers (R). The range is the set of non-negative real numbers, as the output of squaring any real number is always positive or zero. The graph of this function is a parabola. Linear Functions: These are functions of the form f(x) = ax + b or mx + c, where a (or m) and b (or c) are real numbers and a ≠ 0. They define a straight line when graphed. A specific example given is 3.5x + 5.7. Another linear function example is f(x) = 7x + 2 and f(x) = x. Quadratic Functions: These are functions of the form f(x) = ax² + bx + c, where a ≠ 0, and a, b, c are real numbers. The graph of any quadratic function is always a parabola. An example of a shifted parabola is 5x² + 3. Polynomial Functions: A general polynomial function of degree n is described as f(x) = anxⁿ + an-1xⁿ⁻¹ + ... + a₀x⁰, where an ≠ 0 and n is a natural number. They consist of mathematical terms added together, involving only addition, subtraction, multiplication, and natural exponents of variables. An example given is f(x) = x³ + 5. Exponential Functions: These are of the form f(x) = aˣ, where a > 0 and a ≠ 1. The natural exponential function, f(x) = eˣ, is a specific example where e > 1. Other examples include f(x) = 2ˣ and f(x) = (1/2)ˣ. Logarithmic Functions: These are of the form f(x) = logₐ(x), where a > 0 and a ≠ 1. They are the inverse of exponential functions. The natural logarithmic function is f(x) = loge x = ln x, and the common logarithmic function is f(x) = log₁₀ x = log x. The domain is the set of all positive real numbers. Square Root Function: The function f(x) = √x is discussed. By convention, this usually refers to the positive square root. The domain depends on the allowed codomain; if the codomain is restricted to real numbers, the domain is [0, ∞). If complex numbers are allowed as output, the domain can be all real numbers. Absolute Value Function: Denoted by f(x) = |x|, this function returns x if x ≥ 0 and -x if x < 0. It is used as an example to check for injectivity (it is not one-to-one) and continuity at x=0 (it is continuous). Step Functions: Examples include the Floor function, f(x) = ⌊x⌋ (greatest integer value less than or equal to x), and the Ceiling function, f(x) = ⌈x⌉ (smallest integer value greater than or equal to x). Trigonometric Functions: Examples mentioned include sin x, cos x, and tan x. f(x) = sin x is also used to check for differentiability. Constant Function: f(x) = c is used to illustrate differentiation. Rational Function: An example of a real-valued function given is f(x) = (5x+9)/(2x). Function Defined on an Interval: f(x) = 2x - 1 defined on the interval `` is used in the context of calculating area under a curve. Function used in SSE: f(x) = 2x - 2 is implicitly used in a sum squared error calculation example. Bounded Function Example: f(x) = 1/(x² + 1) is shown to be a bounded function with 0 ≤ f(x) ≤ 1. Functions on Other Sets:

🧠

23 Prime NUmbers

Based on the sources and our conversation history, here’s a comprehensive overview of prime numbers: Definition: A prime number is a natural number that has no factors other than 1 and itself. It must have exactly two factors. Factors: The only factors of a prime number p are 1 and p. Why 1 is Not Prime: It is important that a prime number must have two separate factors. While 1 has 1 as a factor (because 1 times 1 is 1), it has only one factor, which is 1 itself. Therefore, 1 is technically not considered a prime number. Smallest Primes: The smallest prime number is 2 because it has exactly two factors: 1 and itself. The next prime numbers are 3, 5, and 7. Even Numbers: After the number 2, no even numbers can be prime because they are all multiples of 2, meaning 2 divides them in addition to 1 and themselves. For example, 4 is divisible by 2, and 6 is not prime because it’s a multiple of 3. Generating Primes (Sieve of Eratosthenes): There is a method called the sieve of Eratosthenes to generate prime numbers. You start by listing numbers (e.g., from 1 to 100). You know 1 is not prime. You take the first unmarked number, which is 2, declare it a prime, and then knock off all its multiples (all the even numbers) as non-primes. Then, you look for the next number that hasn’t been marked off, which is 3, declare it a prime, and mark off all its multiples (some of which might already be marked). You continue this process; the next unmarked number will be the next prime (e.g., 5 is found this way). This method is a good way to generate primes up to a certain number without missing any. Prime Factorization: A very important fact is that every number can be uniquely factorized into the prime numbers that form it. This is also called the prime factorization. For example, 12 can be written as 2 × 6 or 4 × 3, but its fundamental unique prime factorization is 2 × 2 × 3, or using exponentiation, 2² × 3. Similarly, 126 is 2 × 3² × 7. This unique decomposition property is used implicitly a lot. Infinitude of Primes: It is a known result that the set of prime numbers is an infinite set. There cannot be a largest prime number. Euclid provided a proof for this. The proof involves assuming there is a finite list of all primes (p₁, p₂, …, pk), constructing a new number N by multiplying all these primes together and adding 1 (N = p₁ × p₂ × … × pk + 1). This new number N must be larger than any prime in the list. If the list was exhaustive of all primes, N must be composite (not prime). If N is composite, it must have a prime factor, and this prime factor must be in the original list (say pⱼ). So, pⱼ divides N. However, pⱼ also divides the product p₁ × p₂ × … × pk (since pⱼ is one of the factors). A property of divisibility states that if a number divides a sum (a+b) and also divides one part (a), it must divide the other part (b). In this case, pⱼ divides N (the sum) and pⱼ divides the product (one part), so pⱼ must divide 1 (the other part). But pⱼ is a prime number, which is by definition greater than 1, and therefore cannot divide 1. This is a contradiction, meaning the initial assumption (that the list of primes is finite) must be false. Thus, the set of primes is infinite, and there is no largest prime. Distribution: Prime numbers have been extensively studied in an area called number theory. One topic is their distribution within the natural numbers. As numbers get larger, the gaps between primes tend to become larger. The function π(x) denotes the number of primes less than or equal to a given number x. For large x, π(x) is approximately x / log(x). Applications: Despite seemingly abstract, prime numbers are actually quite useful. One important application is in cryptography. Cryptography affects day-to-day life, such as protecting electronic commerce transactions. Much of this encryption relies on the existence of large prime numbers and the fact that it is difficult to factorize the product of two large primes. Computational Problems: There are two related computational problems: checking if a number is prime (primality testing) and finding the prime factors of a number (factorization). Primality testing can be done efficiently. However, there is no efficient way to factorize a large number. This paradox (being able to check if a number is prime efficiently, but not being able to factorize it quickly if it isn’t prime) is why primes are important in cryptography. Set Representation: The set of prime numbers can be defined as a subset of the natural numbers. Using set comprehension, the set of primes (P) can be defined as the set of natural numbers p such that the factors of p consist of exactly two elements {1, p}, and p is not 1. In summary, prime numbers are foundational in number theory, possess unique properties like the basis for unique prime factorization, are infinite in quantity, and have significant practical applications, particularly in securing digital communications.

🧠

24 Why is a number irrational?

Based on the sources and our conversation history, a number is considered irrational primarily because it cannot be written in the form of p/q, where p and q are integers. In simpler terms, irrational numbers are defined as numbers that are not rational. Here are some key points and examples from the sources explaining why numbers are irrational: Definition: The set of rational numbers (Q) consists of numbers that can be expressed as a fraction p/q, where p and q are integers. Irrational numbers are those real numbers that fall outside this definition.

🧠

25 set versus collection

Based on the sources and our conversation history, the key distinction between a set and a collection arises from foundational issues in set theory, particularly when dealing with infinite collections. Here’s a breakdown: Informally, a set is often described as a collection of items. Another source defines a set as a collection of well-defined items. For finite collections, the idea that any collection is a set seems reasonable. Examples of finite sets given include the days of the week, the factors of a number like 24, the set of natural numbers less than 10, and months in a year. However, the sources explicitly state that not every collection is in fact a set. This becomes particularly problematic when dealing with infinite collections. The reason for this caution stems from what is known as Russell’s Paradox. This paradox questioned the fundamental idea that any collection could be considered a set, especially in the context of trying to use set theory as a foundation for all of mathematics. Bertrand Russell’s work demonstrated that the collection of “all sets” leads to a problem if one assumes it is itself a set. The paradox is illustrated by considering a collection, say ‘S’, defined using set comprehension as the subset of all sets that do not contain themselves. If we assume that the collection of all sets exists as a set, then this subset ‘S’ is also a set because it is defined by applying a condition (set comprehension) to an existing set (the set of all sets). The paradox then asks whether this set ‘S’ belongs to itself. If ‘S’ belongs to itself, then by its own definition (sets in S do not contain themselves), it cannot belong to itself. This is a contradiction. If ‘S’ does not belong to itself, then by the condition used to define ‘S’ (sets in S are those that do not contain themselves), it must belong to itself. This is also a contradiction. The conclusion from this paradox is that the initial assumption that leads to the contradiction must be false. The assumption that goes wrong is the assumption that there is a set of all sets. Therefore, not every collection can be called a set; specifically, the collection of all sets does not exist as a set. In modern mathematics, when one is unsure if a collection satisfies the necessary properties to be a set, it is sometimes safer to refer to it as a class. A class is simply a collection of objects that may not have the implied properties expected from sets. The sources advise that when working with sets, especially in this course, it is safe to assume that known collections of numbers like natural numbers (N), integers (Z), rational numbers (Q), and real numbers (R) are sets. New sets should be constructed from these existing, well-defined sets using standard set operations like union, intersection, Cartesian product, and set comprehension. The caution is against “manufacturing sets out of thin air”, such as the problematic “set of all sets”. In summary, while “set” is often used informally to mean “collection,” the sources highlight that in formal mathematics, particularly concerning infinite collections and foundational issues, not every collection is a valid set due to paradoxes like Russell’s Paradox. The collection of all sets is a key example of a collection that is not a set. The term “class” is sometimes used for collections that might not be sets.

🧠

26 Degree of infinity

Based on the sources and our conversation history, the concept of “Degrees of Infinity” refers to the idea that different infinite sets can have different “sizes” or cardinalities. While the term “Degree of Affinity” was not found in the sources, “Degrees of Infinity” was discussed as a way to understand the magnitude of infinite sets. Here’s what the sources explain about this concept: Cardinality is the term used to denote the number of elements in a set. For finite sets, determining cardinality is straightforward – you simply count the elements. This count results in a natural number. The challenge arises when dealing with infinite sets. Examples of infinite sets discussed include the natural numbers (N), the integers (Z), the rational numbers (Q), and the real numbers (R). Some of these sets, like natural numbers and integers, are described as discrete, while rational numbers are described as dense. Despite these structural differences, the question is whether they all have the same “size” or if there are more elements in one infinite set than another. This leads to the core question: are there degrees of infinity?. To compare the sizes of infinite sets, the concept of a bijection is used. If a bijection (a one-to-one and onto function) exists between the set of natural numbers (N) and another set X, it means you can effectively pair up the elements of N with the elements of X. This process allows you to enumerate the elements of X, essentially listing them out in an ordered sequence. Sets that can be enumerated in this way are considered to have the same cardinality as the natural numbers. Examples in the sources include the set of integers and even pairs of integers (Z cross Z), suggesting they can be enumerated. However, the sources demonstrate that the set of real numbers (R) has a higher degree of infinity than the natural numbers. This is shown using a method similar to Cantor’s diagonal argument. By considering infinite sequences of 0s and 1s (which can represent real numbers, specifically those between 0 and 1), it’s proven that no matter how you try to list or enumerate all such sequences, you can always construct a new sequence that is not on your list. This argument shows that the set of infinite 0,1 sequences is not countable. Since these sequences can be mapped to real numbers in the interval, this proves that even this small fraction of the real numbers, and therefore the entire set of real numbers (R), cannot be enumerated. This distinction—being able to enumerate (like N, Z, Q) versus not being able to enumerate (like R)—means that the set of real numbers is “larger” in terms of cardinality than the set of natural numbers, integers, or rational numbers. This demonstrates that there are indeed different degrees of infinity. The sources also briefly mention the continuum hypothesis, a significant open question in set theory concerning whether there exist infinite sets with cardinality strictly between that of the natural numbers (countable infinity) and the real numbers (uncountable infinity). It is noted that this question was shown to be independent of the standard axioms of set theory. In summary, the concept of “degrees of infinity” highlights that not all infinite sets are the same size; sets like the real numbers are proven to be “larger” (uncountable) than sets like the natural numbers (countable), establishing different levels of infinity.

🧠

27 Activity Questions 2.1

Question 1 (Multiple Choice) 1. Choose the correct option with respect to the points P(5, -3), Q(-3, 3), R(0, -100), and S(-2.5, 0) on the rectangular coordinate system. Options: Point R does not lie in any quadrant Points P and R lie in Quadrant III Points S and Q lie in Quadrant II Points R and S cannot be represented on the rectangular coordinate system Solution Question 2 (Multiple Select Questions - MSQ)

🧠

28 area of a triangle

The area of a triangle in a coordinate system is a fundamental concept in geometry that helps us quantify the space enclosed by three non-collinear points 📐. Imagine you have three friends standing at different spots on a flat field, and you want to know how much ground they cover if you connect them with invisible lines to form a triangle 📍📍📍. The area formula provides a precise way to calculate this!

🧠

29 distance formula

The distance formula is a powerful tool in mathematics that helps us measure the shortest straight-line distance between any two points in a rectangular coordinate system. Think of it like a superhero’s tape measure for your graph paper! 📏🦸‍♀️ What is the Distance Formula? The distance formula is built upon the famous Pythagorean Theorem (a² + b² = c²), which applies to right-angled triangles 📐. In a rectangular coordinate system, we can always form a right-angled triangle using the two points and a third auxiliary point, with the distance between the two original points forming the hypotenuse.

🧠

30 Distance of a line from a given point

When considering the distance of a line from a given point in coordinate geometry, we are typically interested in the shortest distance between that point and the line. This shortest distance is always the perpendicular distance from the point to the line. Imagine dropping a perfectly straight plumb line from the point down to the line; that’s the distance we’re calculating! 📏 The General Equation of a Line (Recap) Before diving into the distance formula, let’s quickly recall the general equation of a line, which is: $\mathbf{Ax + By + C = 0}$

🧠

31 equation of a perpendicular line passing through a point

Understanding the equation of a perpendicular line passing through a specific point involves combining your knowledge of the general equation of a line and the conditions for perpendicularity between lines. Let’s break it down! 🧩 The General Equation of a Line (Refresher) 📏 As we discussed, the general equation of a line is given by: $\mathbf{Ax + By + C = 0}$ From this form, if B is not equal to 0, the slope (m) of the line can be found using the formula: $\mathbf{m = -A/B}$. This relationship helps us understand how the constants A and B determine the line’s inclination.

🧠

32 equation of parallel and perpendicular lines in general form

The General Equation of a Line is a powerful way to represent any straight line in the coordinate plane. As we discussed previously, it’s like the master key 🗝️ for lines! The general form of a linear equation is: $\mathbf{Ax + By + C = 0}$ Here, A, B, and C are constant real numbers, and x and y are the variables for the coordinates of any point on the line. It’s crucial that A and B are not both zero simultaneously, otherwise, it wouldn’t represent a line.

🧠

33 General Equation of line

The General Equation of a Line is the most comprehensive and universal way to represent any straight line in the coordinate plane. It’s like the master key 🗝️ that fits all types of lines! Understanding the General Equation of a Line 📏 The general equation of a line is written in the form: $\mathbf{Ax + By + C = 0}$ Here’s what each part means: x and y: These are the variables that represent the coordinates of any point ((x, y)) lying on the line. A, B, and C: These are real number constants. Crucial Condition ⚠️: For this equation to truly represent a line, the constants A and B cannot both be zero simultaneously. If they were, the equation would simplify to C = 0, which is either 0 = 0 (true for all points) or a false statement (e.g., 5 = 0), neither of which describes a single line. Why is it so Universal? 🌐 One of the greatest strengths of the general form is its ability to represent every type of straight line, including those that other forms might struggle with, like vertical lines.

🧠

34 Parallel Lines

The slope of a line tells us about its steepness and direction 📈. Building on our previous discussion about the slope of a line, we can now understand a very important relationship between lines: parallel lines. What are Parallel Lines? Imagine two straight roads that run side-by-side forever and never cross paths, no matter how far they extend 🛣️🛣️. That’s essentially what parallel lines are in geometry!

🧠

35 Rectangular Coordinate System

The rectangular coordinate system is a fundamental concept in mathematics that allows you to precisely locate points on a plane and study geometric objects algebraically. It’s essentially a reference system that gives every point a unique address! 📍 Here’s a breakdown of its key components and applications: What is a Rectangular Coordinate System? A rectangular coordinate system, also known as a Cartesian coordinate system, uses two fixed perpendicular lines called axes to specify the position of any point in a plane. The term “rectangular” comes from the fact that the two axes meet at a 90-degree angle (recta means “right” in Latin).

🧠

36 Representation of a Line - 2

To uniquely represent a line in coordinate geometry, we need more than just its slope, as infinitely many lines can share the same slope. Beyond the horizontal, vertical, point-slope, and two-point forms, there are other powerful ways to define a line uniquely. These forms provide definite conditions or algebraic expressions that describe the line in terms of its coordinates. Let’s explore these additional representations: 1. Slope-Intercept Form 📈📍 This form is widely used due to its direct representation of the line’s steepness and where it crosses the Y-axis.

🧠

37 Representation of a Line 1

Representing a line uniquely in coordinate geometry goes beyond just knowing its steepness or direction (its slope). While a slope tells us how inclined a line is, it doesn’t specify its exact position on the coordinate plane, as there can be infinitely many lines with the same slope. To uniquely represent a line, we need a definite condition or an algebraic expression that describes it in terms of its coordinates.

🧠

38 Section Formula

The section formula is a super handy tool in coordinate geometry! 📐 It helps us find the exact coordinates of a point that divides a line segment connecting two other points, in a specific ratio. Imagine you have a rope stretched between two friends, and you want to put a knot at a certain proportion along the rope – the section formula tells you precisely where that knot would be! 🎀

🧠

39 slope of a line

The slope of a line is a super important concept in coordinate geometry that tells us two main things about a straight line: its direction and its steepness 📈. Think of it like climbing a hill ⛰️: is it going uphill or downhill? And how steep is that climb? The slope tells us exactly that, numerically. What is the Slope of a Line? In simple terms, the slope is a ratio of the change in the vertical direction (y-axis) to the change in the horizontal direction (x-axis). It’s often called “rise over run” 🏃‍♂️⬆️.

🧠

40 Straight line fit

When we talk about a straight line fit in mathematics, especially in the context of data science, we’re essentially trying to find a single straight line that best represents a collection of data points on a graph 📈. Imagine you’ve collected some data from an experiment, and you expect there to be a linear relationship between your measurements, but due to slight errors or variations, the points don’t form a perfect straight line. A “straight line fit” helps you draw that ideal line through the scatter of points! ✨

🧠

41 Activity Questions 2.2

🧠

42 Activity Questions 2.3

Question 1 (Fill in the blank) The coordinates of the midpoint of points P(4, -2) and Q(-1, -1) are _______ Options: (0.5, -0.5) (1.5, -1.5) (-0.5, -0.5) (-0.5, 0.5) Solution Question 2 (Fill in the blank)

🧠

43 Activity Questions 2.4

Multiple Choice Questions (MCQ): Question 1: Choose the correct statement based on the three points P(0, 10), Q(-20, -30) and R(10, 30) Options: The given points form a triangle of area 5 square units The given points form a triangle of area 15 square units The given points do not form a triangle None of the above Solution Question 2: The area of the triangle formed by the midpoints of line segments PQ, QR, and RP where the coordinates of P, Q, and R are (0, 0), (3, 0), and (3, 4) respectively, is ______

🧠

44 Activity Questions 2.5

Multiple Choice Questions (MCQ): Question 1: Find the slope of a line passing through the origin and the point (-3, -2) Options: 1/6 -2/3 -1/3 3/2 Solution Question 2: If the slope of a line passing through P(1, 0) and Q(-2, k) is 1, then the value of k is _____

🧠

45 Activity Questions 2.6

Multiple Choice Questions (MCQ): Question 1: If a line is perpendicular to the X-axis, then the slope of such line is _______ Options: 0 Not defined 1 -1 Solution Question 2: If a line is parallel to a line which is perpendicular to the Y-axis, then the slope of the first line is _______

🧠

46 Activity Questions 2.7

Multiple Choice Questions (MCQ): Question 1: Which of the following represents a equation of the horizontal line? Options: y = 0 x = 5 x = -2 x = 0 Solution Question 2: The equation of a line parallel to the X-axis and passing through the point (-2, 0) is _____

🧠

47 Activity Questions 2.8

Multiple Choice Questions (MCQ): Question 1: The equation of a line passing through (-1, -1) with value of slope 1 is _____ Options: y = -x y = x y = -x - 1 y = -x + 1 Solution Question 2: The equation of a line which cuts the X-axis at (5, 0) and Y-axis at (0, 5) is _____

🧠

48 Activity Questions 2.9

Multiple Choice Questions (MCQ): Question 1: If the x-intercept and the y-intercept of a straight line are -6 and 7 respectively, then choose the correct equation of the line. Options: 7x - 6y + 42 = 0 -6x + 7y - 1 = 0 7x - 6y - 1 = 0 -6x + 7y - 2 = 0 Solution Question 2: The slope of the line 6x - 2y + 8 = 0 is _____

🧠

49 Activity Questions 2.10

Multiple Select Question Question 1: Which of the following statements are true? Options: Lines 2x + 3y - 8 = 0 and 3x - y - 2 = 0 are parallel lines Lines 3x + 5y - 10 = 0 and 6x + 10y - 26 = 0 are parallel lines Lines 6x + 8y - 20 = 0 and 4x - 3y = 0 are perpendicular to each other

🧠

50 Activity Questions 2.11

Multiple Choice Question Question 1: The equation of a line passing through the point (3, 4) and perpendicular to the line 3x + 4y - 8 = 0 is Options: 8x - 6y = 0 2x + 8y = 38 8x + 4y = 5 x + y = 1 Solution Numerical Answer Type

🧠

51 Activity Questions 2.12

Multiple Select Questions Question 1: If the general form of a line is 3x + 2y - 5 = 0, then choose the correct set of options. Options: The slope of the given line is -3/2 The x-intercept is 3 The point where the given line cuts the X-axis is (5/3, 0) The y-intercept is 2 The point where the given line cuts the Y-axis is (0, 5/2) Solution Question 2: Given the point (-2, 1) and the line -3x + 4y - 7 = 0, choose the correct set of options.

🧠

52 Activity Questions 2.13

Numerical Answer Type Question 1: If a line fit y = x + 1 is given for the data as shown in Table AQ-3.1, then compute the Sum Squares Error (SSE). Table AQ-3.1: x 1 2 3 4 6 y 1 1 2 5 7 Solution Multiple Choice Question Question 2: If the relation between x and y is as shown in Table AQ-3.2, then which among the following lines is the best fit?

🧠

53 Activity Questions 3.1

Question 1: Which of the graphs in Figure 1 represents the following function: $y = x^2 - x + 1$? This is a multiple choice question with four options (A, B, C, and D), each showing different graphs. The question asks students to identify which graph correctly represents the given quadratic function.

🧠

54 Quadratic functions

A quadratic equation is a powerful mathematical tool that arises when a quadratic function is set to be equal to a specific value, often zero. Think of it as finding the exact ‘spots’ on a graph where a U-shaped or inverted U-shaped curve (called a parabola) crosses a certain horizontal line. 🎢 What is a Quadratic Equation? At its heart, a quadratic equation is defined by an equation of the form: $\mathbf{ax^2 + bx + c = 0}$

🧠

55 Activity Questions 3.2

Question 1 The curve on the surface of the banana as shown in Figure 2 can be described using the equation $y = x^2 + 2x + 4$. An ant (shown in blue color in Figure 2) is walking from one end of the banana to the other end. What will be the x-coordinate of the ant’s location once it reaches the vertex of its path?

🧠

56 Examples of Quadratic functions

Let’s explore quadratic functions! 🤩 They might sound complex, but they’re just a special kind of equation that helps us understand U-shaped curves. Think of them as the mathematical way to describe things like the path of a thrown ball 🏈, the shape of a bridge arch 🌉, or even how a company’s profit changes with pricing. 📈 What is a Quadratic Function? 🤔 A quadratic function is defined by an equation that looks like this: $\mathbf{f(x) = ax^2 + bx + c}$

🧠

57 Activity Questions 3.3

Question 1 Calculate the slope of the parabola at a point (x, y) obtained by plotting the following function: $y = x^2 + 2x + 4$ Multiple choice options: $2x + 2$ $+2x$ $-1.5x$ $0.5$ Solution Question 2 Calculate the slope of the parabola at a point (x, y) obtained by plotting the following function: $y = -5x^2 + 10x + 10$

🧠

58 Slope of quadratic function

Alright, let’s talk about the slope of a quadratic function! 🎢 Understanding slope is like figuring out how steep a path is at any given moment. For straight lines, the steepness (slope) is always the same, but for curvy paths like parabolas, it changes! 🏔️ What is Slope? (A Quick Refresher) 📏 First, let’s quickly recap what slope means for a linear function (a straight line, like y = mx + c). The slope, often denoted as m, tells you how much the ‘rise’ (vertical change) happens for a certain ‘run’ (horizontal change). It’s calculated as (change in y) / (change in x) or (y2 - y1) / (x2 - x1) for any two points (x1, y1) and (x2, y2) on the line. For a straight line, this m value is constant.

🧠

59 Activity Questions 3.4

Question 1 A stone is thrown with an initial speed u (m/s) as shown in figure 3. The height of the stone’s trajectory above the ground is $H(t) = -5t^2 + \frac{1}{2}ut$ (where t is the time of flight). If the highest point in air that the stone can reach is 5m above the ground, then calculate the initial speed u. The question includes a diagram showing the parabolic trajectory of the stone reaching a maximum height of 5 meters.

🧠

60 Solution of quadratic equation using graph

Right, let’s explore how to solve quadratic equations using their graphs! 📊 This method helps us visually find the answers to quadratic equations by looking at where their curves meet the x-axis. What is a Quadratic Equation and Function? 🤔 A quadratic function is typically expressed in the form f(x) = ax² + bx + c, where a cannot be zero. If a were zero, it would simply be a linear function, which represents a straight line. The graph of any quadratic function is always a U-shaped curve called a parabola.

🧠

61 Activity Questions 3.5

Question 1 The slope of a line which passes through the vertex and the y-intercept of the quadratic equation $x^2 + 10x - 5$ is Multiple choice options: $\sqrt{1230}$ $-5$ $5$ $\sqrt{650}$ Solution Question 2 Identify the point at which the slope of the equation $x^2 + 2x - 5$ is 10

🧠

62 Slope Line & Parabola

Let’s dive into the fascinating world of slopes and parabolas, and how understanding them helps us solve quadratic equations using graphs! 📊 Understanding Slope 🏞️ Imagine you’re walking on a hill. The slope tells you how steep that hill is. In mathematics, it measures how much a line or a curve rises or falls for a given horizontal distance. For a straight line 📏, the slope is constant. This means the steepness never changes. We calculate it using two points on the line, (x1, y1) and (x2, y2), with the formula: m = (y2 - y1) / (x2 - x1)

🧠

63 Activity Questions 3.6 - Summary Lecture

Summary Lecture

🧠

64 Summary Lecture (Quadratic Functions)

Based on the “Summary lecture”, this video summarises the topics covered regarding quadratic equations and functions. The lecture begins by positioning quadratic functions as a generalisation of the concept of a straight line or a linear function, which is typically in the form of mx + b. A quadratic function is defined in the form f(x) = ax² + bx + c, with the crucial condition that a is not equal to 0. If ‘a’ were equal to 0, the term with x² would disappear, and it would simply reduce to a linear function. The name “quadratic” is related to the term “square”.

🧠

65 Activity Questions 3.7

Question 1 Choose the correct standard form of a quadratic equation with roots $\frac{2}{3}$ and $\frac{10}{3}$. Multiple choice options: $6x^2 - x - 70 = 0$ $\frac{1}{3}(6x^2 - x - 70) = 0$ $\frac{1}{6}(6x^2 - x - 7) = 0$ $(x - \frac{2}{3})(x + \frac{10}{3}) = 0$ Solution Question 2 Choose the correct option about a with the help of Figure AQ-5.1.

🧠

66 Solution of quadratic equation using Factorization

Let’s explore how to solve quadratic equations using the factorization method! 🧩 This method is a powerful way to find the “roots” (solutions) of a quadratic equation. What is a Quadratic Equation? 🧐 A quadratic equation is formed when a quadratic function, typically in the form f(x) = ax² + bx + c, is set equal to a specific value, most commonly zero. The standard form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are coefficients and a cannot be zero. If a were zero, it would become a linear equation, not a quadratic one.

🧠

67 Activity Questions 3.8

Multiple Choice Questions (MCQ): Question 1 What should be added in p(x) to make it perfect square, if p(x) = x² - 12x + 34? Multiple choice options: 1 2 3 4 Solution Question 2 Choose the correct option regarding equation x² - 12x + 37 = 0.

🧠

68 Solution of quadratic equation using Square method

Solving a quadratic equation using the completing the square method is a clever way to transform the equation into a form where finding the solutions (or “roots”) becomes straightforward! 🌱 This method is also fundamental to understanding the derivation of the well-known quadratic formula. What is a Quadratic Equation? 🧐 A quadratic equation is typically written in its standard form as ax² + bx + c = 0. Here, a, b, and c are coefficients, and a must not be zero (otherwise, it would simply be a linear equation). The solutions to this equation are called its roots, which are the values of x that make the equation true. Graphically, these roots represent the x-intercepts of the associated quadratic function’s graph (a parabola).

🧠

69 Activity Questions 3.9

Multiple Choice Questions (MCQ): Question 1 The quadratic equation $9x^2 + 6x + 1 = 0$ has Multiple choice options: Two distinct roots Equal roots No roots Inadequate information Solution Question 2 If two different quadratic equations have same discriminant then choose the correct option.

🧠

70 Quadratic formula

Solving a quadratic equation using the Quadratic Formula is a universal method that provides the solutions (or “roots”) for any quadratic equation in standard form. This powerful formula is actually derived directly from the “completing the square” method, making it a reliable tool even when other methods like factoring are difficult or impossible 🛠️. What is a Quadratic Equation? 🧐 A quadratic equation is typically expressed in its standard form as ax² + bx + c = 0. In this equation:

🧠

71 Summary lecture

Understanding quadratic equations and functions is key in mathematics, as they describe many real-world phenomena 🌍. Let’s break down the concepts, formulas, and methods for solving them. What is a Quadratic Function? 🧐 A quadratic function is defined by an equation of the form f(x) = ax² + bx + c, where a is not equal to 0 (a ≠ 0). This condition (a ≠ 0) is crucial because if a were 0, the x² term would disappear, reducing it to a linear function. The term “quadratic” itself is derived from a word meaning “square,” indicating its relation to the square of a variable.

🧠

72 Polynomials

Hello! Let’s explore the fascinating world of polynomials together, with some easy-to-understand explanations and fun emojis! 📚✨ What are Polynomials? From a Layman’s Perspective, a polynomial is simply a mathematical expression that’s a sum of several mathematical terms. Each of these terms can be a number, a variable, or a product of several variables. For example, 3x² + 4y² + 2z + 10 would be considered a polynomial.

🧠

73 Degree of Polynomials

Hello there! Let’s dive into understanding the degree of polynomials with some fun and easy-to-digest explanations, complete with emojis and practice questions! 🎓✨ What is a Polynomial? 🤔 From a “Layman’s perspective,” a polynomial is a mathematical expression that’s essentially a sum of several mathematical terms. Each of these “mathematical terms” can be a number, a variable, or a product of several variables. However, a “mathematician’s perspective” gives us a more precise definition: a polynomial is an algebraic expression where the only arithmetic operations allowed are addition, subtraction, multiplication, and variables can only have “natural exponents”. “Natural exponents” mean non-negative integers (0, 1, 2, and so on). For instance, an expression like “t raised to half plus t” (t^(1/2) + t) would not qualify as a polynomial because t^(1/2) has a rational exponent, not a natural one.

🧠

74 Algebra of Polynomials

Hello there! Great to continue our discussion on polynomials! You’re keen to understand the Algebra of Polynomials, specifically Addition and Subtraction, with ease and emojis. Let’s get right to it! ➕➖ What is a Polynomial? (A Quick Recap! 🔄) Before we add and subtract, let’s briefly recall what a polynomial is. From a “Layman’s perspective,” a polynomial is a mathematical expression that is essentially a sum of several mathematical terms. Each of these “mathematical terms” can be a number, a variable, or a product of several variables. For instance, 3x is a term, x²y is a term, and 10 is a term.

🧠

75 Polynomial Multiplication!

Alright! Let’s dive into the fascinating world of Polynomial Multiplication! ✨ We’ve already covered addition and subtraction, which are about combining “like terms”. Multiplication introduces a new twist, but it’s just as logical and easy to grasp. Get ready for some algebra fun! 🚀 What is a Polynomial Again? (A quick mental stretch! 🧠) Just to quickly recap from our previous conversation, a polynomial is a mathematical expression that’s essentially a sum of several mathematical terms. Each of these terms can be a number, a variable, or a product of several variables, like 3x or x²y. From a mathematician’s perspective, the operations allowed are addition, subtraction, multiplication, and variables must have “natural exponents” (non-negative integers like 0, 1, 2, etc.).

🧠

76 Polynomials Division

Right then! We’ve mastered the art of adding and subtracting polynomials, and last time, we unravelled the secrets of multiplication. Now, let’s tackle the final frontier of polynomial algebra: Division! ➗ It’s a bit like long division with numbers, but with variables thrown into the mix. Don’t worry, we’ll make it as easy as pie (polynomial pie, of course! 🥧). What is Polynomial Division? 🤔 Just like when you divide numbers (e.g., $10 \div 3$ gives $3$ with a remainder of $1$), polynomial division involves splitting one polynomial (the dividend) by another (the divisor) to find a quotient and a remainder.

🧠

77 Division Algorithm

Alright, let’s dive into the exciting world of Polynomial Division! ➗ It’s the final major arithmetic operation for polynomials, and while it might seem a bit daunting at first, it’s very much like the long division you already know, just with some algebraic twists! 😉 What is Polynomial Division? 🤔 Just as you can divide whole numbers, for example, $10 \div 3$ results in $3$ with a remainder of $1$, you can divide one polynomial by another. This process aims to break down a more complex polynomial (the dividend) into simpler parts using another polynomial (the divisor).

🧠

78 Polynomial Functions from a Graph 🕵️‍♀️

Understanding the graphs of polynomial functions involves a twofold mission: first, being able to identify whether a given graph represents a polynomial function, and second, understanding the key characteristics that shape the graph of a polynomial. Identifying Polynomial Functions from a Graph 🕵️‍♀️ When presented with a graph, you can determine if it’s a polynomial function by checking for two main properties: Smoothness ✨: Polynomial functions always display smooth curves, meaning they do not have any sharp corners or edges. If you try to draw a polynomial graph, you should be able to join the points effortlessly without experiencing any “abrupt jerk”. If a graph has a corner or an edge, it is unlikely to be a polynomial function. For example, graphs of linear and quadratic functions (which are types of polynomials) are drawn smoothly without jerks. Continuity 〰️: Polynomial functions are continuous curves, meaning they have no breaks. You should be able to draw the entire graph without lifting your pen. If a graph requires you to lift your pen to continue drawing (indicating a break or discontinuity), then it is not a polynomial function. For instance, a graph that is smooth but has a sharp corner, like one shown in the sources, would not qualify as a polynomial function. Similarly, a graph that is smooth in sections but has a break where you’d need to lift your pen, like the one illustrated with a discontinuity at x=0, would also be disqualified. Conversely, a graph that is visibly smooth and continuous, resembling a line or a curve with gentle turns, would qualify as a polynomial function.

🧠

79 Identifying Zeros of Polynomials

Understanding the zeros of a polynomial function is like finding the special points where the graph of the function crosses or touches the horizontal axis (the x-axis) 🎯. These points are also known as x-intercepts. For a polynomial function, f(x), a value of x is a zero if f(x) = 0. Let’s break down how to identify and characterise these zeros from a graph or an equation. Identifying Zeros of Polynomials 🕵️‍♀️ When you’re trying to find the zeros of a polynomial, you’re essentially looking for the x-values that make the function’s output zero. Here’s how you can find them:

🧠

80 Graphs of Polynomials Multiplication

Understanding the graphs of polynomial multiplication involves seeing how the characteristics of individual polynomial graphs combine to determine the characteristics of their product’s graph. While the sources do not provide a direct method for visually multiplying graphs without algebraic computation, they offer a comprehensive explanation of how to multiply polynomials algebraically and then characterise the resulting graph. Here’s a breakdown of polynomial multiplication and its graphical implications: 1. Algebraic Polynomial Multiplication ✖️ At its core, multiplying polynomials is a process of term-by-term multiplication. You multiply each term of the first polynomial by every term of the second polynomial and then combine “like terms” (terms with the same variable and exponent). The law of exponents is applied, meaning you add the exponents of the variables when multiplying. Any constant coefficients are multiplied throughout the expression.

🧠

81 Graphs of Polynomaials | Turning Point

Let’s explore turning points in the graphs of polynomial functions! 🎢 A turning point on a polynomial graph is a specific location where the graph changes its direction. Imagine you’re tracing the graph with your finger: If your finger was moving upwards (the function was increasing 📈) and now it starts moving downwards (the function is decreasing 📉), that point is a turning point. This is called a local maximum. Conversely, if your finger was moving downwards (the function was decreasing 📉) and now it starts moving upwards (the function is increasing 📈), that point is also a turning point. This is called a local minimum. These “ups and downs” are typical features of polynomial functions. You can visualise them as the peaks and valleys on the curve.

🧠

82 Graphs of Polynomials | Graphing & Polynomial creation

Let’s delve into the fascinating world of graphs of polynomials, focusing on how to sketch them and even create their equations from a given graph! 🎢 Polynomial functions are special types of functions that are always smooth curves with no sharp corners or edges. They are also continuous, meaning you can draw their entire graph without lifting your pen. The “ups and downs” you see in their graphs are typical features.

🧠

83 Graphs of Polynomials Behavior at X-intercepts

Let’s explore how the graph of a polynomial behaves when it touches or crosses the x-axis, which is where its “zeros” or “x-intercepts” are found! 🧐 Understanding Zeros and Multiplicities 🎯 First, what are zeros of a polynomial? They are simply the values of ‘x’ for which the polynomial function ‘f(x)’ equals zero. Graphically, these are the points where the graph crosses or touches the x-axis. When a polynomial is written in its factored form, such as f(x) = (x - a)ᵐ, the number ‘a’ is a zero of the polynomial. The exponent ’m’ in this factor is called the multiplicity of that zero. The multiplicity is essentially how often that factor is appearing. The behaviour of the graph at each x-intercept is critically determined by this multiplicity.

🧠

84 Graphs of Polynomials End Behavior

Let’s delve into the end behaviour of polynomial graphs! This describes what happens to the graph of a polynomial function as the x values become very large (approaching positive infinity, x → ∞) or very small (approaching negative infinity, x → -∞). The Role of the Leading Term 🚀 The end behaviour of a polynomial is determined solely by its leading term. The leading term is the term with the highest degree (highest exponent) in the polynomial. For very large or very small values of x, this term will dominate and essentially dictate the overall direction of the graph, making all other terms insignificant in comparison.

🧠

85 One to One Functions | Definition & Tests

A one-to-one function, also known as an injective function, is a type of mathematical relationship where each unique input from the domain maps to a unique output in the co-domain. Think of it like assigning a unique identifier to every item: if two items have the same identifier, they must be the exact same item. 🎯 More formally, a function f: A → B is considered one-to-one if for any two elements x₁ and x₂ in the domain A:

🧠

86 One to One Functions | Definition & Tests

Let’s explore Exponential Functions in an easy-to-understand way! 🚀 What is an Exponential Function? 🤔 An exponential function is a mathematical function that shows rapid growth or decay. It’s defined with a constant base raised to a variable exponent. Formally, an exponential function in standard form is described as: f(x) = a^x Where: a is the base. a must be greater than 0 (a > 0). a cannot be equal to 1 (a ≠ 1). x is the variable exponent. Think of it like compound interest, where your money grows (or shrinks) at an accelerating rate! 💰📈📉

🧠

87 One to One Functions | Definition & Tests

Let’s dive into Graphing Exponential Functions! 📈📉 In our previous discussion, we established that an exponential function is generally defined as f(x) = a^x, where the base a is a positive constant (a > 0) and not equal to one (a ≠ 1). Now, let’s explore how these functions look when plotted and what their key characteristics are. Core Characteristics of Exponential Graphs (f(x) = a^x) 📊 Regardless of the specific value of the base a (as long as it meets the definition criteria), all standard exponential functions f(x) = a^x share some fundamental graphical properties:

🧠

88 One to One Functions | Definition & Tests

Let’s explore Natural Exponential Functions! 🌿📈 The natural exponential function is a special type of exponential function where the base is the mathematical constant e. This e is approximately 2.71828. Because e is greater than 1 (e > 1), the natural exponential function behaves like the “growth” type of exponential function we discussed earlier. Definition and Key Characteristics of f(x) = e^x 📊 The natural exponential function is defined as f(x) = e^x. It shares many fundamental graphical properties with other exponential functions of the form f(x) = a^x where a > 1:

🧠

89 One to One Functions | Definition & Tests

Let’s dive into Composite Functions! 🔗✨ Imagine you have two machines 🤖. One machine (g) takes an input and gives an output. Then, you take that output and feed it into a second machine (f), which then gives you a final output. That’s exactly what a composite function is! It’s when the output of one function becomes the input of another function. What is a Composite Function? 🤔 A composite function is essentially a function inside another function. If you have two functions, say f and g, the composition of f and g is written as f ◦ g. This is defined by (f ◦ g)(x) = f(g(x)).

🧠

90 One to One Functions | Definition & Tests

Absolutely! Let’s dive into one-to-one functions, making it easy to understand with definitions, tests, examples, and practice questions. 🎯 What is a One-to-One Function? 🤔 A one-to-one function, also known as an injective function, is a special type of mathematical relationship where each unique input from the domain maps to a unique output in the co-domain. Think of it like this: Imagine a class where every student has a unique student ID. No two different students can have the same ID. That’s a one-to-one relationship! ✅🧑‍🎓🆔 If, however, two different students could have the same ID (e.g., student 1 and student 2 both have ID 123), then it’s not one-to-one. 🙅‍♀️ More formally, a function f: A → B is considered one-to-one if for any two elements x₁ and x₂ in the domain A:

🧠

91 Composite Functions | Examples

Let’s make understanding Composite Functions as easy as pie! 🥧✨ Imagine you have a couple of magical machines 🤖⚙️. Machine G (the inner function g): Takes your initial idea (x) and transforms it into something new (g(x)). Machine F (the outer function f): Takes what Machine G made (g(x)) and transforms that into a final product (f(g(x))). That’s precisely what a composite function is! It’s when the output of one function becomes the input of another function 🔗.

🧠

92 Composite Functions | Domain

Alright, let’s dive into composite functions again, this time focusing on their domain with some friendly emojis! 🤖✨ What are Composite Functions? 🤔 As we discussed, imagine two magical machines 🤖⚙️: Machine G (the inner function g): Takes your initial idea (x) and transforms it into something new (g(x)). Machine F (the outer function f): Takes what Machine G made (g(x)) and transforms that into a final product (f(g(x))). A composite function is essentially a “function of a function” [Conversation History]. If you have two functions, f and g, their composition is typically written as f ◦ g (read as “f of g”), and it’s formally defined by the equation:

🧠

93 Inverse Functions

Let’s unravel the world of inverse functions! 🔄✨ What are Inverse Functions? 🤔 Imagine a magical function machine f 🤖 that takes an input x and spits out an output f(x). An inverse function, denoted as f⁻¹ (read as “f inverse”), is like a reverse magic machine 🪄. Its job is to undo what the original f machine did. If you put f(x) into f⁻¹, it will give you back the original x! [5.10]