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12th Maths_Functions 2_Avanti Module

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M17 – Functions 2 1 Fifth Edition M17. Functions 2 TABLE OF CONTENTS M17. Functions 2 1 M17.1 Types of Relations and Functions........................................................................................................................................2 M17.2 Composite Functions, Inverse Functions and Binary Operations .........................................................................9 M17.3 Graphical Transformation and Periodicity...................................................................................................................13 M17.4 Problem Solving Techniques (Advanced Topic)........................................................................................................23 Test Practice Problems .........................................................................................................................................................................25 Answer Key ................................................................................................................................................................................................29

M17 – Functions 2 2 M17.1 Types of Relations and Functions CONCEPTS 1. Identify whether a given relation is reflexive, symmetric or transitive. 2. Classify functions as one – one, many – one, onto and into. 3. Identify nature of functions, i.e. whether a function is even, odd or neither even nor odd. PRE-READING Category Book Name Chapter Section 1 1.1, 1.2, 1.3 REQUIRED NCERT Class XII Mathematics – Part 1 ADDITIONAL PRE-READING 1. Even function If a function ������ = ������(������) satisfies the condition ������(−������) = ������(������) for all values of ������, then ������ = ������(������) is called an even function. Note: As an even function satisfies ������(−������) = ������(������), ������(������) possesses same value for values of ������ which are equal in magnitude and opposite in sign. For example: ������(−1) = ������(1), ������(−2) = ������(2), ������(−3) = ������(3) … Therefore, graph of an even function is symmetrical about ������-axis ,i.e., left half is mirror image of right half and right half is mirror image of left half, considering ������-axis as a mirror. Illustrating the concept: I. Consider ������(������) = ������2 ������(−������) = (−������)2 = ������2 ⇒ ������(−������) = ������(������) Hence ������(������) = ������2 is an even function. Note: Graph of ������(������) = ������2 is symmetrical about y-axis. II. Consider ������(������) = cos ������ ������(−������) = cos(−������) = cos ������ [Using cos(−������) = cos ������] ⇒ ������(−������) = ������(������) Hence, ������(������) = cos ������ is an even function. Note: Graph of ������(������) = cos ������ is symmetrical about y-axis. M17.1

M17 – Functions 2 3 2. Odd Function If a function ������ = ������(������) satisfies ������(−������) = −������(������) for all values of ������, then ������ = ������(������) is called an odd function. Note: As an odd function satisfies ������(������) = −������(������), ������(������) possesses values equal in magnitude but opposite in sign for all values of ������ which are also equal in magnitude but with opposite signs. For example: ������(−1) = −������(1), ������(−2) = −������(2), ������(−3) = −������(3) … Therefore, graph of an odd function is symmetrical about origin, i.e., if we rotate the right half of the graph anticlockwise by 180° about origin then we get graph in left half. Illustrating the concept: I. Consider ������(������) = ������3 ������(−������) = (−������)3 = −������3 = −������(������) ⇒ ������(������) satisfies ������(−������) = −������(������) Hence ������(������) = ������3 is an odd function. II. Consider ������(������) = sin ������ ������(−������) = sin(−������) = − sin ������ = −������(������) [Using sin(−������) = −sin(������) ] ⇒ ������(������) satisfies ������(−������) = −������(������) Hence, ������(������) = sin ������ is an odd function. Note: As graph is symmetrical about origin, left half of the graph can also be drawn by taking reflection of right half in both x-axis as well as y-axis: 3. Properties of Even and Odd Functions Let there are two real valued functions ������: ������ → ������ and ������: ������ → ������. Then ������(������) ������(������) ������(������) ± ������(������) ������(������) × ������(������) ������(������) ÷ ������(������) EVEN EVEN EVEN EVEN EVEN EVEN ODD NEITHER ODD ODD ODD ODD ODD EVEN EVEN ODD EVEN NEITHER ODD ODD I. a) If ������(������) + ������(−������) = 0 ⇒ ������(������) is an odd function b) If ������(������) − ������(−������) = 0 ⇒ ������(������) is an even function. II. The first derivative of an odd function is an even function and first derivative of an even function is an odd function. III. The square of an even or an odd function is always an even function. IV. Any function ������ = ������(������) can be written as a sum of an even and an odd function, i.e., ������ = ������(������) = ������(������) + ������(������) + ������(−������) − ������(−������) ⇒ ������(������) = [������(������)+������(−������)] + [������(������)−������(−������)] 22 2 2 22 where [������(������)+������(−������)] is an even part of ������(������) and [������(������)−������(−������)] is an odd part of ������(������). 22

M17 – Functions 2 4 SYNOPSIS 1. TYPES OF RELATIONS : a VOID RELATION : A relation ������ in a set ������ is called the empty or void relation, if no element of ������ is related to any element of ������. As, ������ ⊂ ������ × ������, for any set ������, so ������ is an empty relation on ������. b UNIVERSAL RELATION : A relation ������ in a set ������ is called the universal relation, if each element of ������ is related to every element of ������. Since, ������ × ������ ⊆ ������ × ������, so ������ × ������ is universal relation on ������. c IDENTITY RELATION : The relation, ������������ = {(������, ������) ∶ ������ ∈ ������} is called the identity relation on ������. d REFLEXIVE RELATION : A relation ������ is said to be reflexive, if every element of ������ is related to itself. Thus, (������, ������) ∈ ������, ∀ ������ ∈ ������ ⇒ ������ is reflexive. A reflexive relation can have other terms besides (������, ������), while identity will only have (������, ������). e SYMMETRIC RELATION : A relation R is said to be symmetric, iff (������, ������) ∈ ������ and (������, ������) ∈ ������, ∀ ������, ������ ∈ ������ i.e., ������������������ ⇒ ������������������, ∀ ������, ������ ∈ ������ ⇒ ������ is symmetric. f TRANSITIVE RELATION : A relation ������ is said to be transitive, iff (������, ������) ∈ ������ and (������, ������) ∈ ������ ⇒ (������, ������) ∈ ������, ∀ ������, ������, ������ ∈ ������ g EQUIVALENCE RELATION : A relation ������ is said to be an equivalence relation, if it is simultaneously reflexive, symmetric & transitive on ������. 2. INVERSE RELATION : If ������ and ������ are two non-empty sets and ������ be a relation from ������ to ������, such that ������ = {(������, ������) ∶ ������ ∈ ������, ������ ∈ ������}, then the inverse of ������, denoted by ������−1 , is a relation from ������ to ������ and is defined by ������−1 = {(������, ������) ∶ (������, ������) ∈ ������}. 3. POINTS TO REMEMBER : a) Both the empty relation and the universal relation are sometimes called trivial relations. b) If ������ and ������ are two equivalence relations on a set ������, then ������ ∩ ������ is also an ‘equivalence relation on ������’. c) The union of two equivalence relations on a set is not necessarily an equivalence relation on the set. d) If ������ is an equivalence relation on a set ������, then ������−1 is also an equivalence relation on ������. e) If a set ������ has ������ elements, then number of reflexive relations from ������ to ������ is 2������2−������. f) Let ������ and ������ be two non-empty finite sets consisting of ������ and ������ elements, respectively. Then, ������ × ������ consists of ������������ ordered pairs. So, total number of relations from ������ to ������ is 2������������. 4. FUNCTION : ������: ������ → ������ ⇒ ������ is a function from set ������ to set ������, if to each element ������ ∈ ������, there exists a unique element ������ ∈ ������. 5. DOMAIN AND RANGE OF A FUNCTION : a) DOMAIN : All possible values of ������ for which ������(������) exists. b) RANGE : All possible values of ������(������), for all values of ������. i.e., ������������ = {������ ∈ ������: ������ = ������(������)} c) ������������ ⊆ Co-domain M17.1

M17 – Functions 2 5 6. ONE-ONE FUNCTION OR INJECTION: A function ������: ������ → ������ is one-one iff a) ������ ≠ ������ ⇒ ������(������) ≠ ������(������) b) ������(������) = ������(������) ⇒ ������ = ������ Here, ������ = {������, ������, ������, ������} and ������ = {������, ������, ������, ������, ������, ������} 7. ONTO FUNCTION OR SURJECTION: A function ������: ������ → ������ is onto iff Range of ������ = Co-domain of ������. Here, ������ = {������, ������, ������, ������} and ������ = {������, ������, ������} 8. INTO FUNCTION: A function ������: ������ → ������ is an into function, if there exists an element in ������ having no pre-image in ������. Here, ������ = {������, ������, ������} and ������ = {������, ������, ������, ������, ������} 9. MANY-ONE FUNCTION: ������: ������ → ������ is a many-one function, if ∃������, ������ ∈ ������ such that ������ ≠ ������, but ������(������) = ������(������). Here, ������ = {������, ������, ������, ������} and ������ = {������, ������, ������, ������, ������, ������}

M17 – Functions 2 6 10. BIJECTIVE FUNCTION: A function both injective and surjective is called bijective function. Here, ������ = {������, ������, ������, ������} and ������ = {������, ������, ������, ������} PRE-READING EXERCISE Q1. Let ������ = {1, 2, 3, 5}. The relation on the set ������ given by ������ = {(1, 1), (2, 2), (3, 5)} is reflexive. True or False? Q2. Let ������ be the relation in the set ������ given by ������ = {(������, ������): ������ = ������ – 2, ������ > 6}. Choose the correct answer. A) (2, 4) ∈ ������ B) (3, 8) ∈ ������ C) (6, 8) ∈ ������ D) (8, 7) ∈ ������ Q3. If a function is both one-one and onto, then it is called a __________ function. Q4. If range of a function is same as the codomain, then it is an __________ function. Q5. Using vertical line test, check whether this graph represents a function or not? IN CLASS EXERCISE 1 Q1. Let ������ be a relation on the set ������ of natural numbers defined by ������������������ if ������ divides ������ exactly (i.e. zero remainder). Then ������ is A) Reflexive and symmetric B) Transitive and symmetric C) Equivalence D) Reflexive and transitive Q2. Let ������ denote the set of all straight lines in a plane. Let a relation ������ be defined by ������������������ if and only if ������ is perpendicular to ������ ∀ ������, ������ ∈ ������. Then ������ is A) Reflexive B) Symmetric C) Transitive D) None of these Q3. If the functions ������ and ������, given by ������(������) = log(������ − 1) − log(������ − 2) and ������(������) = log (������−1) are equal, then ������ lies in ������−2 the interval A) [1, 2] B) (2, ∞) C) [2, ∞) D) (−∞, ∞) M17.1

M17 – Functions 2 7 IN CLASS EXERCISE 2 Q4. *This diagram illustrates that the function shown is (* indicates multiple options correct) A) One–one B) Onto C) Many-one D) Into Q5. Let a function is defined, ������: ������ → ������ as ������(������) = 7. Then ������(������) is A) One–one but not onto B) onto but not one – one C) Neither one – one nor onto D) Both one – one and onto Q6. The function ������: ������ → ������ defined by ������(������) = −������2 + 6������ − 8 is a bijective function, if A) ������ = (−∞, 3] and ������ = (−∞, 1] B) ������ = [3, ∞) and ������ = ������ C) ������ = (−∞, 3] and ������ = [1, ∞) D) ������ = [3, ∞) and ������ = [1, ∞) Q7. Show that all the rational functions of the form ������(������) = 1 where ������ and ������ are real numbers such that ������ ≠ 0 and ������������+������ ������ ≠ − ������ are one – one functions. ������ Q8. Let ������ = {1, 2, 3, 4} and ������ = {1, 2}. Then the number of onto functions from ������ to ������ is A) 14 B) 16 C) 6 D) 4 IN CLASS EXERCISE 3 Q9. Determine whether the following functions are even, odd, or neither. I. ������(������) = 1−������2 II. ������(������) = (������2 + ������)2 III. ℎ(������) = |������| 1+������2 ������+������7 Q10. Determine whether the given function, ℎ(������) = ������3 − ������2 − 1 is even, odd or neither. Also, rewrite the function as a sum of an even and an odd function. Q11. Write the function ������(������) = 2������3 + 9 sin2 ������ + 5������ as a sum of an even and an odd function. HOMEWORK LEVEL 1 Q1. Determine whether each of the following relations are reflexive, symmetric and transitive: I. Relation ������ in the set ������ = {1, 2, 3, … 13, 14} defined as ������ = {(������, ������): 3������ − ������ = 0} II. Relation ������ in the set ������ of natural numbers defined as ������ = {(������, ������): ������ = ������ + 5 and ������ < 4} III. Relation ������ in the set ������ = {1, 2, 3, … 13, 14} as ������ = {(������, ������): ������ is divisible by ������} IV. Relation ������ in the set ������ of human beings in a town at a particular time given by a) ������ = {(������, ������): ������ is the wife of ������} b) ������ = {(������, ������): ������ is the father of ������} Q2. Find whether the function ������(������) = 3√������(������3 − ������) is even, odd or neither even nor odd. Q3. Which diagram below represents a function that is onto, but not one - one? A) 1 B) 2 C) 3 D) None

M17 – Functions 2 8 Q4. The relation from set ������ to set ������ in the given figure is B) ������(������) = ������3 + 4 from ������ to ������ D) ������(������) = ������2 + 4 from ������ to ������ A) Neither one - one nor onto function B) Injective (one - one), but not a surjective function C) Surjective (onto), but not an injective function D) Bijective (both one - one and onto) function Q5. Which of the following functions is NOT injective? A) ������(������) = ������3 + 4 from ������ to ������ C) ������(������) = ������2 + 4 from ������ to ������ LEVEL 2 Q6. Show that the relation ������ in the set ������ of all the books in a library of a college, given by ������ = {(������, ������): ������ and ������ have same number of pages} is an equivalence relation. Q7. Prove that the “greatest integer function” ������: ������ → ������ given by ������(������) = [������], is neither one - one nor onto, where [������] denotes the greatest integer less than or equal to ������. Q8. If ������ is a one - one function defined on the interval(−5, −2) ∪ (−2, 5), then find the number of values of ������ satisfying ������(������) = ������ (������+1). ������+2 Q9. Show that the relation ������ defined in the set ������ of all polygons as ������ = {(������1, ������2): ������1 and ������2 have same number of sides} is an equivalence relation. What is the set of all elements in ������ related to the right – angled triangle ������ with sides 3, 4 and 5? LEVEL 3 Q10. ������(������) = (sin ������7 )e(������5 ������������������(������)9) is A) Odd B) Even C) Neither D) Constant Q11. Show that the function ������: ������∗ → ������∗ defined by ������(������) = 1 is one-one and onto, where ������∗ is the set of all non – zero ������ real numbers. Is the result true, if the domain ������∗ is replaced by ������ with co-domain being same as ������∗? M17.1

M17 – Functions 2 9 M17.2 Composite Functions, Inverse Functions and Binary Operations CONCEPTS 1. Compute composite of two or more functions. 2. Identify whether the inverse of a function exists and to find the inverse of a function. 3. Identify whether a function is a binary operation or not, and check for commutativity and associativity of binary operations. 4. Find identity and inverse for binary operations. PRE-READING Category Book Name (Edition) Chapter Section 1 1.4, 1.5 REQUIRED NCERT Class XII Mathematics – Part 1 ADDITIONAL PRE-READING An important property of the inverse functions is : The graph of a function and its inverse function are always the mirror image of each other in the line ������ = ������. Let us understand this concept with the help of an example. Q. Find the inverse of the function ������(������) = ������2 − 1 and also draw the graphs of ������(������) and ������−1(������) (limit the domain of the function to the interval in which the function is increasing) Soln. The given function ������(������) = ������2 − 1 is a parabola opening upwards with its vertex at (–������ , −������), i.e., (0, −1) 2������ 4������ ⇒ the function is increasing after ������ = 0, so the domain is (0, ∞) To find the equation of the inverse, We need to solve the equation ������ = ������2 − 1 for ������. ⇒ ������ = ������2 − 1 ⇒ ������ + 1 = ������2 ⇒ ������ = ±√(������ + 1) ⇒ ������ = √(������ + 1) [∵ domain is (0, ∞)] ⇒ ������−1(������) = √(������ + 1) Now, we will make the graphs by making table of values: ������ ������(������) ������−1(������) −1 0 0 0 −1 1 1 0 √2 2 3 √3 382 4 15 √5 The two graphs are shown next to the table. We can see that the two graphs are mirror images of each other in the line ������ = ������. Also each point on the graph of ������(������) is mirror image of a point on ������−1(������). For example, if (√2, 1) is on ������(������), then (1, √2) is on ������−1(������). In general, if (������, ������) is on ������(������), then (������, ������) is on ������−1(������).

M17 – Functions 2 10 SYNOPSIS 1. INVERSE OF A FUNCTION: If ������: ������ → ������ be a one-one, onto function, then the mapping ������−1: ������ → ������ such that ������−1(������) = ������ is called inverse of the function ������: ������ → ������. 2. COMPOSITION OF FUNCTIONS : Let ������: ������ → ������ and ������: ������ → ������, then ������������������: ������ → ������, such that: (������������������)(������) = ������{������(������)}, ∀������ ∈ ������ 3. BINARY OPERATIONS : Let ������ be a non-empty set. A function ������ from ������ × ������ to ������ is called a binary operation on ������ i.e., ������ ∶ ������ × ������ → ������ is a binary operation on set ������. Generally binary operations are represented by the symbols *, +, … etc., instead of letters figure etc. 4. CLOSURE PROPERTY : An operation * on a non-empty set ������ is said to satisfy closure property, iff ������ ∈ ������, ������ ∈ ������ ⇒ ������ ∗ ������ ∈ ������, ∀ ������, ������ ∈ ������. 5. ASSOCIATIVE LAW : A binary operation * on a non-empty set ������ is said to be associative, if (������ ∗ ������) ∗ ������ = ������ ∗ (������ ∗ ������), ∀ ������, ������, ������ ∈ ������. 6. COMMUTATIVE LAW : A binary operation * on a non-empty set ������ is said to be commutative, if ������ ∗ ������ = ������ ∗ ������, ∀ ������, ������ ∈ ������. 7. DISTRIBUTIVE LAW : Let * and ������ be two binary operations on a non-empty sets. Then * is distributed over ������., if ������ ∗ (������ ������ ������) = (������ ∗ ������) ������ (������ ∗ ������), ∀ ������, ������, ������ ∈ ������ also called (left distribution) and (������ ������ ������) ∗ ������ = (������ ∗ ������) ������ (������ ∗ ������), ∀ ������, ������, ������ ∈ ������ also called (right distribution) 8. IDENTITY ELEMENT : Let * be a binary operation on a non-empty set ������. An element ������ in a non-empty set ������, if it exists such that ������ ∗ ������ = ������ ∗ ������ = ������, ∀ ������ ∈ ������, is called an identity element of S, with respect to *. 9. INVERSE OF AN ELEMENT : Let * be a binary operation on a non-empty set ‘������’ and let ‘������’ be the identity element. Let ������ ∈ ������, if there exists an element ������ ∈ ������ such that ������ ∗ ������ = ������ ∗ ������ = ������ then ������ is called the inverse of ������. Addition on ������ has no identity element and accordingly ������ has no invertible element. Multiplication on ������ has 1 as the identity element and no element other than 1 is invertible. PRE-READING EXERCISE Q1. If ������(������) = sin ������ and ������(������) = ������2, then ������������������(������) = ������������������(������). True or false? Q2. The inverse of ������(������) exists if it is A) One – one and into B) One – one and onto C) Many – one and onto D) Many – one and into Q3. If the function ������(������) is inverse of ������(������), then ������������������(������) = __________. Q4. The operation ÷ : ������ × ������ → ������ is not a binary operation. True or false? Q5. The property ������ × ������ = ������ × ������ is known as __________. Q6. Given ������(2) = 3, ������(3) = 2, evaluate ������������������(3). Q7. Find the value of ������������������−1 (2.3). M17.2

M17 – Functions 2 11 IN CLASS EXERCISE 1 Q1. Find domain and range of the following functions: I. ������(������) = ln(3������2 + 2) II. ������(������) = ln (3������21+2) III. ������(������(������)) if ������(������) = sin(������) and ������(������) = 1 ������−1 Q2. ������(������) is a piecewise function defined by: ������(������) = {���������2���,, ������ < 0 ������ ≥ 0 Function ������(������) is defined by ������(������) = √������. Find the composite function ������(������(������)). Q3. Evaluate ������ (������(ℎ(1))). Given that ℎ(������) = −|������|, ������(������) = ������ − 1 and (������) = 1 . (������+4) Q4. The number ������ of bacteria in food is given by: ������(������) = 20������2 – 80������ + 800, such that 2 ≤ ������ ≤ 14,where ������ is the temperature of the food in oC. When the food is removed from refrigerator, the temperature of the food is given by ������(������) = 4������ + 2, such that 0 ≤ ������ ≤ 3, where ������ is time in hours. Find the composition ������(������(������)) and time when the bacteria count reaches 2000. Q5. Find domain and range of ������������������(������) if ������(������) = ln(������) and ������(������) = ������2 − 3������ + 2. IN CLASS EXERCISE 2 Q6. Prove that linear functions are bijective functions and further prove that the straight lines in the graph shown below are inverses of one another. (The ������ and ������ intercepts of ������1 are −5 and 2.5 respectively; ������ and ������ interecpts of ������2 are 2.5 and −5 respectively) Q7. Let ������ = {1, 2, 3}. Determine whether the functions ������: ������ → ������ defined as below have inverses. Find ������−1, if it exists. I. ������ = {(1,1), (2,2), (3,3)} II. ������ = {(1,2), (2,1), (3,1)} III. ������ = {(1,3), (3,2), (2,1)} Q8. Let ������: ������ → ������ be a function defined as ������(������) = 4������ + 3, where, ������ = {������(������) ∀ ������ ∈ ������}. Show that ������ is invertible. Find the inverse. IN CLASS EXERCISE 3 Q9. Consider the binary operation ∧ on the set {1, 2, 3, 4, 5} defined by ������ ∧ ������ = min{������, ������}. Write the operation table of the operation ∧.

M17 – Functions 2 12 Q10. Find whether the following operation ∗ on ������ = {0, 1} is commutative or not ∗ 01 010 111 Q11. Let a binary operation be defined such that ������ ∗ ������ = 2������ + ������, find (2 ∗ 3) ∗ 5 & 2 ∗ (3 ∗ 5). Is the operation associative? Q12. Determine whether the binary operation ∗ given by ������ ∗ ������ = ������+������ is a binary operation on the set of integers. ������2 HOMEWORK LEVEL 1 Q1. Let ������: ������ → ������ and ������: ������ → ������ be two real valued functions such that ������(������) = 3������ + ������, ������(������) = ������−4. Find value(s) of ������ 3 for which ������������������ = ������������������. Q2. The natural logarithmic function is inverse function of the exponential function. Since the point (0, 1) lies on the graph of exponential function, we know that the point __________ lies on the graph of logarithmic function. A) (0, 1) B) (1, 1) C) (1, 0) D) (−1, 1) Q3. Let ∗ be a binary operation defined on the set of natural numbers ������ = {1, 2, 3 … }, given by ������ ∗ ������ = ������������. Is ∗ commutative? Is ∗ associative? LEVEL 2 Q4. Are the following tables commutative for ∗ binary operation? Why or why not? (������, ������, ������ are distinct real numbers) I. ������ ������ ������ II. ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ Q5. If functions ������ and ������ are given by ������(������) = √������ + 2 and ������(������) = ln(1 − ������2). Then find the composite function defined by (������������������)(������) and describe its domain. Q6. If ������(������) = (4������+3) , ������ ≠ 2, show that ������������������(������) = ������, for all ������ ≠ 2. What is the inverse of ������? (6������−4) 3 3 Q7. The value of parameter ������, for which the function ������: ������ → ������ given by ������(������) = 1 + ������������, ������ ≠ 0 is the inverse of itself, is A) −2 B) −1 C) 1 D) 2 Q8. If ������(������) = ������2, ������(������) = 2������, ℎ(������) = ������ − 2, find ((������������������)������ℎ)(������) and (������������(������������ℎ))(������). Comment on the associativity of the composite functions thus found. Q9. Find the domain of following functions: II. ������������������(������) if ������(������) = tan(������) and ������(������) = ln(������) I. ������(������(������)) if ������(������) = sin(������) and ������(������) = 1 ������−1 LEVEL 3 Q10. Consider ������: [0, ∞) → [4, ∞) given by ������(������) = ������2 + 4 . Show that ������ is invertible with the inverse ������−1 given by ������−1(������) = √������ − 4. 1 ������ > 0 Q11. Let ������: ������ → ������ be a signum function defined as ������(������) = { 0 ������ = 0 and ������: ������ → ������ be the greatest integer function −1 ������ < 0 given by ������(������) = [������], where [������] is greatest integer less than or equal to ������. Then do ������������������ & ������������������ coincide in (0,1]? M17.2

M17 – Functions 2 13 M17.3 Graphical Transformation and Periodicity CONCEPTS 1. Determine the graphs of ������(������) ± ������, ������(������ ± ������), ������������(������), ������(������������), |������(������)| and ������(|������|) given the graph of ������(������). 2. Determine the periodicity and fundamental period of standard functions. 3. Use graphical transformations to determine the period of ������(������) ± ������, ������(������ ± ������), ������������(������), ������(������������) ), |������(������)| and ������(|������|) given the fundamental period of ������(������). PRE-READING 1. Transformation of Curves If we are given a graph of a standard function such as sin ������ , ������������, ln ������ etc., then using that particular graph, we can move and resize the graph of the function into a new one using few typical operations. Let us start with a function, ������(������) = ������2. The graph of ������(������) = ������2 is shown as: Note: In the following graphs, dotted line curves are of ������(������) and solid line curves are of ������(������). Some simple operations that we can do on this graph are as follows: I. We can shift it upward or downward by adding a constant ������ to ������2. Therefore, the equation becomes ������(������) + ������ = ������2 + ������ Let ������(������) + ������ = ������(������) ⇒ ������(������) = ������2 + ������ Note: Graph shown above is for ������ > 0. ������ > 0 shifts the graph upward. ������ < 0 shifts the graph downward. II. We can shift the graph leftward or rightward by adding a constant ������ to the ������ – value. In other words replace ������ by ������ + ������. Therefore, the equation becomes ������(������ + ������) = (������ + ������)2 for ������(������) = ������2. Let ������(������) = ������(������ + ������) ⇒ ������(������) = (������ + ������)2

M17 – Functions 2 14 By adding ������ to ������, the curve shifts to left side (the negative ������ – direction) for ������ > 0. Why? Because when we are adding a constant to the input variable ������, the input variable is getting increased because of which the graph give those values which we are supposed to get at a later stage in case of ������(������). So to cover all those values the graph has to shift leftward. For instance, At ������ = −������, ������(������ + ������) will give same output as ������(������) at ������ = 0. Imagine you will inherit a fortune when your age = 25. If you change that to (age + 4) = 25 then you would get it when you are 21. Adding 4 made it happen earlier. ������ > 0 shifts the graph leftward. ������ < 0 shifts the graph rightward. Example: The function ������(������) = ������3 − ������2 + 4������ shifts ������ (������ > 0) spaces to the leftward i.e. ������(������) = ������(������ + ������) = (������ + ������)3 − (������ + ������)2 + 4(������ + ������). III. We can stretch or compress the curve in ������ – direction by multiplying the whole function by a constant ������. Therefore, the equation becomes ������. ������(������) = ������. ������2 Let ������(������) = ������. ������(������) ⇒ ������(������) = ������. ������2 ������ > 1 stretches the graph in the ������ − direction. 0 < ������ < 1 compresses the graph in the ������ − direction. IV. We can stretch or compress the curve in the ������ – direction by multiplying ������ (wherever it appears) by a constant ������. Therefore, the equation becomes ������(������. ������) = (������. ������)2 Let ������(������) = ������(������. ������) ⇒ ������(������) = (������. ������)2 ������ > 1 compresses the graph in the ������ − direction. 0 < ������ < 1 stretches the graph in the ������ − direction. Note that unlike for ������ – direction, bigger values of ������ cause more compression. V. We can flip the curve upside down by multiplying the whole function by (−1). Therefore, the equation becomes – ������(������) = −������2 M17.3

M17 – Functions 2 15 Let ������(������) = −������(������) ⇒ ������(������) = −������2 This is also called reflection about the ������ – axis . We can combine a negative value with a scaling. Example: Multiplying by (−2) will flip it upside down and stretch it in the ������ – direction. VI. We can flip the curve left – right by multiplying the ������ value by (−1). Therefore, the equation becomes ������(−������) = (−������)2. Let ������(������) = ������(−������) ⇒ ������(������) = (−������)2 It flips the curve leftward and rightward which can’t be seen because ������2 is symmetrical about the ������ – axis. So here is another example using √������. Therefore, the equation becomes ������(−������) = √−������. Let ������(������) = ������(−������) ⇒ ������(������) = √−������ This is also called reflection about the ������ – axis. SUMMARY TABLE FOR GRAPH TRANSFORMATION ������ = ������(������) + ������ ������ > 0 shifts the graph upward. ������ < 0 shifts the graph downward. ������ = ������(������ + ������) ������ > 0 shifts the graph leftward. ������ < 0 shifts the graph rightward. ������ = ������. ������(������) ������ > 1 stretches the graph in ������ – direction. 0 < ������ < 1 compresses the graph. ������ = ������(������������) ������ > 1 compresses the graph in ������ – direction. ������ = −������(������) 0 < ������ < 1 stretches the graph. ������ = ������(−������) Reflects the graph about ������ – axis. Reflects the graph about ������ – axis. Examples: The curve of the function ������(������) = 1 ������  Shifts 2 spaces up when ℎ(������) = 1 + 2 ������

M17 – Functions 2 16  shifts 3 spaces down when ℎ(������) = 1 − 3 ������  shifts 4 spaces to the right when ℎ(������) = 1 ������−4  shifts 5 spaces to the left when ℎ(������) = 1 ������+5  Stretches by a factor of 2 in the ������ – direction when ℎ(������) = 2 ������  Compresses by a factor of 3 in the ������ – direction when ℎ(������) = 1 3������  Flip it upside down when ℎ(������) = − 1 ������ Multiple Graphical Transformations: More than one transformations can be performed in cases like ������(������(������������ + ������)) + ������. Steps for Multiple Transformations Use the following order strictly to plot the graph of a function involving more than one transformation: a. Do the horizontal translation first, i.e., draw the graph of ������(������ + ������) which means the graph of ������(������) will shift leftward (if ������ > 0) or rightward (if ������ < 0). b. Now, draw the graph of ������(������������ + ������) by stretching the graph of ������(������ + ������) in the horizontal direction, if 0 < ������ < 1 or by compressing the graph of ������(������ + ������) in the horizontal direction, if ������ > 1. If ������ < 0 then take reflection of the graph of ������(������ + ������) about ������-axis. c. Now, draw the graph of ������������(������������ + ������) by compressing the graph of ������(������������ + ������) in the vertical direction if 0 < ������ < 1 or stretching the graph of ������(������������ + ������) in the vertical direction if ������ > 1 and reflection about ������-axis if ������ < 0. d. Finally, do vertical translation, i.e., draw the graph of ������������(������������ + ������) + ������ which means the graph of ������������(������������ + ������) will shift upward (if ������ > 0) or downward (if ������ < 0). Example: ������(������) = −2√������ + 3 + 1 Let √������ is the basic function (Basic function is the function with which we start the graphical transformation) Basic function (b.f.) = √������. Comparing ������(������) = −2√������ + 3 + 1 with ������(������(������������ + ������)) + ������, we get ������ = −2, ������ = 1, ������ = 3, ������ = 1. M17.3

M17 – Functions 2 17 I. Now to draw the graph of √������ + 3 , replace ������ by ������ + 3 in √������. The resultant graph will shift leftward in ������- direction by 3 units. (Horizontal translation) II. Next, multiply √������ + 3 by 2 units to get the graph of 2√������ + 3 which will stretch in ������-direction by 2 units. (stretching in ������-direction) III. Now, multiplying 2√������ + 3 by −1, the graph of 2√������ + 3 will get reflected along ������-axis. (Reflection along ������- axis)

M17 – Functions 2 18 IV. Finally, to draw the graph of −2√������ + 3 + 1, add 1 in −2√������ + 3. The resultant graph will shift upward in ������- direction by 1 unit. (Vertical translation) 2. Periodic function: A function ������(������) is said to be a periodic function of ������, if there exists a positive real number ������ such that ������(������ + ������) = ������(������). The smallest value of ������ is called fundamental period of the function. Note: The value of ������ should be independent of ������ for ������(������) to be periodic. In case ������ is not independent of ������, ������(������) is not a periodic function. Definition (Graphically) A function is said to be periodic if its graph repeats itself after a fixed interval and the width of that interval is called fundamental period of the function. For example: ∵ Graph of ������(������) = sin ������ repeats after an interval of 2������. Thus, ������(������) = sin ������ is periodic with period 2������. M17.3

M17 – Functions 2 19 Standard Results on periodic functions: Fundamental Period Functions ������, when ������ is even 2������, when ������ is odd or a fraction sin������ ������ , cos������ ������ , sec������ ������ , cosec������ ������ ������, when ������ is either even or odd ������ tan������ ������ , cot������ ������ 1 |sin ������|, |cos ������|, |tan ������|, |cot ������|, |sec ������|, |cosec ������| ������ − [������] = {������} Properties of Periodic Functions: I. If ������(������) has period ������, then a. ������������(������) is periodic with period ������, because when ������ is multiplied to ������(������) the range of the function will get amplified ������ times but the period will remain unaffected. b. ������(������ ± ������) is periodic with period ������, because the curve will shift either leftward or rightward but the period of the function will not change. c. ������(������) ± ������ is periodic with period ������, because the curve will shift either upward or downward but the period of the function will not change. d. Every constant function is always periodic, with no fundamental period. e. If ������(������) is periodic with period ������, then ������������(������������ + ������) has period ������ , hence period is affected by coefficient of |������| ������ only. II. If ������(������) and ������(������) are two functions with period ������1 and ������2 respectively and ℎ(������) = ������������(������) + ������������(������) then ℎ(������) has period = LCM of {������1, ������2}. It only works if both ������1 and ������2 are either rational or irrational. Example: Let ������(������) = sin(������) and ������(������) = tan ������ are two periodic functions with their respective fundamental periods as ������1 = 2������ and ������2 = ������. Then the fundamental period of ℎ(������) = ������(������) + ������(������) = sin(������) + tan(������) = LCM of {2������, ������} = 2������ Note: There are some exceptions to above result: For example: Period of ������(������) = |sin ������| + |cos ������| = ������ instead of ������. 2 Period of ������(������) = sin4 ������ + cos4 ������ = ������ instead of ������. 2 Period of ������(������) = |tan ������| + |cot ������| = ������ instead of ������. 2 PRE-READING EXERCISE Q1. Find the period of following functions: I. ������(������) = sin(2������) II. ������(������) = 2������3 + 3 III. ������(������) = 1003cos(������) IV. ������(������) = 2tan ������ + 43 Q2. Given ������(������) = ln(������). From the curve of ������(������) draw the curves of the following transformations: I. ������(������) + 3 II. ������(������ − 3) III. 3������(������) IV. ������(3������)

M17 – Functions 2 20 IN CLASS EXERCISE 1 Q1. Given ������(������) = sin(������). From the graph of ������(������) draw the following graphs. I. ������(������ − 2) II. |������(������ − 2)| III. ������(|������|) Q2. Draw the graphs of II. ������2 − 5|������| + 6 IV. ������ = −|(������ + 1)3| I. ������ = |������2 − 5������ + 6| III. ������ = |������3| Q3. Given the graph of ������(������) below, match the following four functions with their graphs. Function Graph I. 2������(������) 1) II. 1 ������(������) 2) 3) 2 4) III. ������(2������) IV. ������ (������) 2 M17.3

M17 – Functions 2 21 IN CLASS EXERCISE 2 Q4. The graph of a periodic function is shown below. What is the period of the function? Q5. Find the period of the following functions: I. sin 2������������ + cos ������������ 32 II. cos(cos ������) + cos(sin ������) Q6. Find the period of the following periodic function and write down the piece-wise function for the interval ������ ∈ [0, ������] where ������ is the period of the given function. Q7. Sketch the graph of the following periodic functions showing all relevant values: ������2 0 ≤ ������ < 4 4 ≤ ������ < 6 I. ������(������) = { 2 6 ≤ ������ < 8 8 0 ������(������ + 8) = ������(������) II. ������(������) = 2������ − ������2, 0 < ������ < 2, ������(������ + 2) = ������(������) Q8. If ������(������) is an odd function with period 2, then find the value of ������(4). HOMEWORK LEVEL 1 Q1. Find the period of ������(������) = tan 3������ + cos 5������. 2 Q2. The period of the function ������(������) = ������ − [������] is _________ .

M17 – Functions 2 22 LEVEL 2 Q3. Order the following waves according to the length of their periods, from least to greatest. Q4. The graph of a periodic function ������ is shown below. Find: I. The fundamental period of ������ II. ������(98) Q5. If ������(������) = sin ((√[������]) ������) has period 2������, then find the set of possible value(s) of ������. ([������] denotes greatest integer function). Q6. Find the period of the function ������(������) = ������ sin ������������ + ������ cos ������������. Q7. The function ������(������) = sin (������������) − cos ( ������������ ) is B) Periodic with period 2(������!) D) periodic with period 2((������ + 1)!) ������! (������+1)! A) Non periodic C) periodic with period (������ + 1) Q8. Is cos √������ a periodic function? If yes, then find its period. If not, then give reasons to explain your answer. Q9. An automated robot transports building materials between two locations, which are 5 km apart. It takes the robot two hours to deliver the materials and two hours to return. Neglecting loading and unloading time, draw a displacement versus time graph that shows the robot performing three complete trips. From graph, also find period. LEVEL 3 Q10. If the function ������(������) = sin ������ + cos ������������ is periodic, then prove that ������ is a rational number. Q11. The period of the function ������(������) = 4 sin4 (4���6���−������32������) + 2 cos (4���3���−������32������) is A) 3������2 B) 3������3 C) 4������2 D) 4������3 4 4 3 3 Q12. Find period of the function, ������(������) = sin4 ������ + cos4 ������ M17.3

M17 – Functions 2 23 M17.4 Problem Solving Techniques (Advanced Topic) CONCEPTS 1. Find the number of roots of a function by using the concept of intersection of two or more different curves. 2. Solve the functional equations of the type ������(������) + ������(������ − ������), ������(������ + 1) − ������(������) = ������, ������(������)������(������) = ������(������) + ������(������), ������(������������) = ������(������) + ������(������) And ������(������ + ������) = ������(������)������(������). PRE-READING Suppose we are to find the total number of solutions of an equation ������(������) = ������(������). One of the methods would be to normally solve the equation to get the roots and hence, count the number of roots. However, we are not asked to find the roots of the equation but the number of solutions. Thus, graphical approach is better as well as faster. Approach: Take ������ = ������(������) = ������(������) ∴ ������ = ������(������), ������ = ������(������) Now draw the graphs of ������ = ������(������) and ������ = ������(������) and check at how many points the graphs of ������(������) and ������(������) intersect. So, the required total number of solutions will be equal to the total number of points of intersection. IN CLASS EXERCISE 1 Q1. If ������(������ + 1) = 2������(������)+1, ������ = 1, 2 … and ������(1) = 2 then ������(101) equals: 2 A) 52 B) 100 C) 101 D) 51 D) ������2 − 2 Q2. If 2������(������ − 1) − ������ (1−������) = ������, then ������(������) is : ������ A) 1 [2(1 + ������) + 1 ] B) 2(������ − 1) − 1−������ 3 1+������ ������ C) ������2 + 1 + 3 D) None of these ������2 Q3. If ������(������) + 2������(1 − ������) = ������2 + 2, ∀ ������ ∈ ������, then ������(������) is given as: A) (������−1)2 B) (������−2)2 C) ������2 − 1 3 3 Q4. A function ������ satisfies the condition ������ (������ + 1) = ������2 + 1 , ������ ≠ 0. Determine ������(������). ������2 ������ Q5. I. If ������: ������ → ������ is given by ������(������) = 4������ ∀ ������ ∈ ������ . Then prove that ������(������) + ������(1 − ������) = 1. Also find the value of 4������+2 ������ ( 1 ) + ������ ( 2 ) + ⋯ + ������ (98) 99 99 99 II. Let ������: ������ → ������ be a function satisfying the following conditions. a) ������(1) = 1 and b) ������(1) + 2������(2) + ⋯ ������������(������) = ������(������ + 1)������(������) for ������ ≥ 2 Prove that ������(49) = 1 . 98

M17 – Functions 2 24 IN CLASS EXERCISE 2 Find the number of solutions of the following equations: Q6. ������������ = −������ Q7. 2 − |������| = 3|������| Q8. ������������ − ������2 + 2 = 0 Q9. |������| − sin ������ = 0 3 HOMEWORK Q1. If ������, ������ are two fixed positive integers such that 1 ������(������ + ������) = ������ + [������3 + 1 − 3������2 ������(������) + 3������{������(������)}2 − {������(������)}3]3 ∀ ������ ∈ ������ then ������(������) is a periodic function with period: A) ������ B) 2������ C) ������ D) 2������ Q2. If a function ������(������) satisfies the equation ������(������ + 1) + ������(������ − 1) = √3������(������) ∀������ ∈ ������ . Show that ������(������) is a periodic function of period 12. Q3. If ������(������) = cos(log ������), then evaluate ������(������)������(������) − 1 [������ (������) + ������(������������)]. 2 ������ Q4. A cubic function ������(������) satisfies the condition ������(������) + ������ (1) = ������(������)������ (1), then prove that ������(������) = 1 + ������3 or 1 − ������3. ������ ������ And if ������(3) = 28, then find ������(2). Q5. Let ������(������) be a polynomial function satisfying ������(������)������(������) = ������(������) + ������(������) + ������(������������) − 2 ∀ ������, ������ ∈ ������. If ������(2) = 5, then find ������(5). Q6. If for non-zero ������, ������������(������) + ������������ (1) = 1 − 5 where ������ ≠ ������, then find ������(������). ������ ������ Q7. Find the number of solutions of the following equations: I. ������������ = ������4 II. ln ������ = ������2 − 14������ + 48 III. sin ������ = ln ������ IV. sin ������ = ������3 − 3������2 + 2 Q8. Total number of solutions of 2������ + 3������ + 4������ − 5������ = 0 is: A) 0 B) 1 C) 2 D) infinitely many Q9. Number of solution(s) of the system of the equation |������2 − 2������| + ������ = 1, ������2 + |������| = 1 is/are: A) 2 B) 1 C) 3 D) 4 M17.4

M17 – Functions 2 25 Test Practice Problems No. of questions: 25 Total time: 75 Minutes Time per question: 3 Minutes Purpose: To practice a mixed bag of questions in a speed based format similar to what you will face in entrance examinations. In most entrance examinations, you will get not more than 3 minutes to attempt a question. Hence, you need to be able to attempt a question in less than 3 minutes, and at the end of 3 minutes skip the question and move to the next one. Approach:  Attempt the Test Practice Problems only when you have the stipulated time available at a stretch.  Start a timer and attempt the section as a test.  DO NOT look at the answer key / solutions after each question.  DO NOT guess a question if you do not know it. Competitive examinations have negative marking.  Fill the table at the end of the TPP and evaluate the number of attempts, and accuracy of attempts, which will help you evaluate your preparedness level for the chapter. Q1. The graph of a function is given. Is the function even, odd, or neither? A) Even B) Odd C) Neither Q2. The periodic function ������(������) = 2 cos(3������) − 1 has a fundamental period of A) 2������ B) ������ C) − ������ D) 2������ 3 3 3 Q3. Let ������: ������ − {0} ⟶ ������ be defined by ������(������) = 1. Then ������ is ������ A) One-one but not onto B) Onto but not one-one C) Bijective D) ������ is not defined −1 ������ < 0 Q4. Let ������(������) = 1 + ������ − [������] and ������(������) = { 0 ������ = 0 . Then for all ������, ������(������(������)) is equal to 1 ������ > 0 A) 0 B) 1 C) −1 D) 2 Q5. Determine algebraically whether the function ������(������) = −3������4 − ������2 is even, odd, or neither. A) Even B) Odd C) Neither Q6. Consider the non-empty set consisting of children in a family and a relation ������ defined as ������������������ if ������ is brother of ������. Then ������ is A) Symmetric but not transitive B) transitive but not symmetric C) neither symmetric nor transitive D) Both symmetric and transitive

M17 – Functions 2 26 Q7. Let ������ = {1, 2, 3} and consider the relation ������ = {(1,1), (2,2), (3,3), (1,2), (2,3)}. Then ������ is A) Reflexive only B) Reflexive and transitive C) Symmetric and transitive D) Symmetric only Q8. Let ������(������) = ������������ , ������ ≠ −1. Then, for what value of ������ is ������(������(������)) = ������? ������+1 A) √2 B) −√2 C) 1 D) −1 Q9. The identity element for the binary operation ∗ defined on ������ − {0} as ������ ∗ ������ = ������������ is 2 A) 1 B) 0 C) 2 D) None of these Q10. Let ������ = {������ ∈ ������: ������ ≤ 1} and ������: ������ → ������ be defined as ������(������) = ������(2 − ������). Then, ������−1(������) is A) 1 + √1 − ������ B) 1 − √1 − ������ C) √1 − ������ D) 1 ± √1 − ������ Q11. Let function ������: ������ → ������ be defined by ������(������) = 2������ + sin ������. Then ������ is A) One - one and onto B) One - one but NOT onto C) Onto but NOT one - one D) Neither one - one nor onto Q12. Given ������(������) = log (1+������) and ������(������) = 3������+������3 , then ������������������(������) equals 1+3������2 1−������ A) −������(������) B) 3������(������) C) [������(������)]3 D) none of these Q13. Which of the following functions from ������ into ������ are bijections? (������ is the set of all integers) A) ������(������) = ������3 B) ������(������) = ������ + 2 C) ������(������) = 2������ + 1 D) ������(������) = ������2 + 1 Q14. If ������(������) = ������ ������ + ������ and ������(������) = ������ ������ + ������, then ������(������(������)) = ������(������(������)) is equivalent to A) ������(������) = ������(������) B) ������(������) = ������(������) C) ������(������) = ������(������) D) ������(������) = ������(������) Q15. If ������: [0, ∞) → [1, ∞) is defined by ������(������) = ������2 + 1, then the value of ������−1(17) is: A) 3 B) 4 C) 5 D) 6 Q16. Let ∗ be a binary operation defined on ������ by ������ ∗ ������ = 1 + ������������. Then the operation ∗ is A) commutative but not associative B) associative but not commutative C) neither commutative nor associative D) both commutative and associative Q17. Let ������: ������ − {3} → ������ be defined by ������(������) = 3������+2, then: 5 5������−3 A) ������−1(������) = ������(������) B) ������−1(������) = −������(������) C) (������ ������ ������)������ = −������ D) ������−1(������) = 1 ������(������) 19 Q18. Let ������: [1, ∞) ⟶ [1, ∞) be a function defined by ������(������) = (������ − 1)2 + 1 Statement I: The set {������: ������(������) = ������−1(������)} = {1, 2} Statement II: ������ is a bijection and ������−1(������) = 1 + √������ − 1, ������ ≥ 1. A) Both Statement I and Statement II are true and Statement II is a correct explanation of Statement I. B) Both Statement I and Statement II are true but Statement II is not a correct explanation of Statement I. C) Statement I is true but Statement II is false. D) Statement I is false but Statement II is true. T.P.P.

M17 – Functions 2 27 Q19. The set of values of parameter ������ for which the function ������: ������ → ������ defined by ������(������) = ������������ + sin ������ is bijective, is A) [−1, 1] B) ������ − (−1, 1) C) ������ − [−1, 1] D) (−1, 1) Q20. If ������: [1, ∞) → [2, ∞) is given by ������(������) = ������ + 1 then ������−1(������) equals: ������ A) ������+√������2−4 B) ������ C) ������√������2−4 D) ������ + √������2 − 4 1+������2 2 2 Q21. let ������: ������ → ������ be defined by ������(������) = ������ ������ 2 −������ −������ 2 , then: ������ ������ 2 +������ −������ 2 A) ������(������) is one - one but not onto B) ������(������) is neither one - one nor onto C) ������(������) is onto but not one - one D) ������(������) is one - one and onto Q22. If ������(������) = sin2 ������ , ������(������) = √������ and ℎ(������) = cos−1 ������ , 0 ≤ ������ ≤ 1, then A) ℎ������������������������ = ������������������������ℎ B) ������������������������ℎ = ������������ℎ������g C) ������������ℎ������������ = ℎ������������������������ D) None of these Q23. If ������(������) = 1, ������(������) = 1 and ℎ(������) = ������2, then ������2 ������ A) ������������������(������) = ������2 where ������ ≠ 0 and ℎ(������(������)) = 1 ������2 B) ℎ(������(������)) = 1 where ������ ≠ 0 and ������������������(������) = ������2 ������2 C) ������������������(������) = ������2 where ������ ≠ 0 and ℎ(������(������)) = (������(������))2, ������ ≠ 0 D) None of these Q24. If ������(������) is an invertible function and ������(������) = 2������(������) + 5 , then the value of ������−1(������) is: A) 2������−1(������) − 5 B) 1 C) 1 ������−1(������) + 5 D) ������−1 (������−5) 2������−1(������)+5 2 2 Q25. The range of ������(������) = √(1 − cos ������)√(1 − cos ������)√(1 − cos ������)√… ∞ is : A) (0, 1) B) [1, 2] C) (0, 1] D) [0, 2]

M17 – Functions 2 28 DATA ANALYSIS Guide A # of questions Total problems in TPP B # Attempts Total attempts in OMR C # Correct Total questions correct D # Incorrect Out of the ones marked in OMR E # Unattempted ������ − ������ F Percentage attempts ������ ������ × 100 G Percentage Accuracy ������ ������ × 100 Question type # Correct (C) # Incorrect (I) # Unattempted (U) Easy Medium Hard Tip: To begin with, your accuracy must be high, typically > 60%. Percentage attempts should be > 50% As time progresses, your percentage attempts should increase without a reduction in accuracy. Additionally, you should be able to get > 80% Easy questions correct, as they involve basic recall of the concepts and formulae of the chapter. T.P.P.

M17 – Functions 2 29 Answer Key M17.1 TYPES OF RELATIONS AND FUNCTIONS PRE-READING EXERCISE HOMEWORK Q1. False LEVEL 1 Q2. C Q3. Bijective Q1. I. ������ is neither reflexive, nor symmetric, nor Q4. Onto transitive. Q5. No II. ������ is neither reflexive, nor symmetric, nor IN CLASS EXERCISE 1 transitive. Q1. D III. ������ is reflexive and transitive but not Q2. B symmetric. Q3. B IV. a) ������ is neither reflexive, nor symmetric, IN CLASS EXERCISE 2 but ������ is transitive. Q4. B, C b) ������ is neither reflexive, nor symmetric, Q5. C nor transitive. Q6. A Q7. Proof Q2. Even Q8. A Q3. C Q4. A IN CLASS EXERCISE 3 and Q5. C Q9. I. Even LEVEL 2 II. Neither III. Odd Q6. Proof Q7. Proof Q10. Neither. Even part of ℎ(������) = (−������2 − 1) Q8. 2 odd part of ℎ(������) = ������3 Q9. Proof. The set of all elements in ������ related to triangle Q11. ������������(������) = 9 sin2 ������; ������������(������) = 2������3 + 5������ ������ is the set of all triangles. LEVEL 3 Q10. A Q11. Proof. Not true. M17.2 COMPOSITE FUNCTIONS, INVERSE FUNCTIONS AND BINARY OPERATIONS PRE-READING EXERCISE IN CLASS EXERCISE 1 Q1. False Q1. S. No. Domain Range Q2. B Q3. ������ I. ������ ∈ ������ [ln(2) , ∞) Q4. True Q5. Commutative property. II. ������ ∈ ������ (−∞, ln (1)] Q6. (������������������)(3) = 3 Q7. 2.3 2 III. ������ ∈ ������ − {1} [−1, 1] Q2. ������ for ������ ≥ 0 Q3. 0.5

M17 – Functions 2 30 Q4. ������(������(������)) = 320������2 + 720 and ������ = 2 hours, where Q12. ∗ is not a binary operation on integers. 0 ≤ ������ ≤ 3 HOMEWORK Q5. Domain is ������ ∈ (−∞, 1) ∪ (2, ∞) and Range is ������. LEVEL 1 Q1. ������ = 4 IN CLASS EXERCISE 2 Q2. C Q6. Proof Q3. Neither commutative nor associative Q7. I. ������−1 = ������ LEVEL 2 II. ������ is not invertible Q4. I. No III. ������−1 = {(3, 1), (2, 3), (1, 2)} II. Yes Q8. ������(������) = ������−1(������) = ������−3 Q5. (������������������)(������) = ln(−������ − 1) where ������ ∈ [−2, −1) Q6. Inverse of ������ is ������ itself. 4 Q7. B Q8. (������������������)������ℎ(������) = 4(������ − 2)2 IN CLASS EXERCISE 3 Q9. ∧ 1 2 3 4 5 And (������������(������������ℎ))(������) = 4(������ − 2)2 Compositions are associative. 111111 Q9. I. ������ ∈ ������ − {2������������ + ������} where ������ ∈ ������ 212222 312333 2 412344 512345 II. ������ ∈ ������+ − {������(2������+1)���2���} where ������ ∈ ������ Q10. Not commutative LEVEL 3 Q11. (2 ∗ 3) ∗ 5 = 19 and 2 ∗ (3 ∗ 5) = 15 ⇒ The Q10. Proof Q11. No operation is not associative M17.3 GRAPHICAL TRANSFORMATION AND PERIODICITY PRE-READING EXERCISE Q1. I. ������ II. Not Defined III. 2π IV. ������ Q2. I. II. III. IV. Ans.

M17 – Functions 2 31 IN CLASS EXERCISE 1 Q1. I. II. IV. II. III. Q2. I. III. Q3. I. 3 II. 4 III. 1 IV. 2

M17 – Functions 2 32 IN CLASS EXERCISE 2 0 < ������ < 3 Q4. 4 3 ≤ ������ < 5 Q5. I. 12 II. ������ 2 Q6. Period is 5 and definition is ������(������) = {2−−1������ Q7. I. II. Q8. 0 Q9. 4 HOMEWORK LEVEL 1 Q1. 4������ Q2. 1 LEVEL 2 LEVEL 3 Q3. Wave B < Wave C < Wave A Q10. Proof Q11. B Q4. I. 4 Q12. ������ II. 0 2 Q5. 1 ≤ ������ < 2 Q6. 2������ ������ Q7. D Q8. No because period depends on ������ which violates definition of the period of a function. M17.4 PROBLEM SOLVING TECHNIQUES (ADVANCED TOPIC) IN CLASS EXERCISE 1 Q5. I. 49 II. Proof Q1. A Q2. A IN CLASS EXERCISE 2 Q3. B Q6. 1 Q4. ������(������) = ������2 − 2 Q7. 2 Ans.

M17 – Functions 2 33 Q8. 1 Q5. ������(5) = 26 Q9. 2 Q6. ������(������) = 1 [������ − ������������] − 5 HOMEWORK ������2−������2 (������+������) Q1. B ������ Q2. Proof Q3. 0 Q7. I. 3 Q4. ������(2) = 9 II. 2 III. 1 IV. 3 Q8. B Q9. C TEST PRACTICE PROBLEMS Q. No. Ans. Level Mark (C) / (I) / Q. No. Ans. Level Mark (C) / (I) / (U) as appropriate (U) as appropriate Easy Easy Q1. A Medium Q14. C Medium Q2. D Medium Q15. B Easy Q3. A Medium Q16. A Medium Q4. B Easy Q17. A Hard Q5. A Hard Q18. A Hard Q6. B Easy Q19. C Hard Q7. A Medium Q20. A Hard Q8. D Easy Q21. B Medium Q9. C Medium Q22. D Easy Q10. B Medium Q23. C Medium Q11. A Medium Q24. D Medium Q12. B Medium Q25. D Q13. B


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