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1 | 718-721 | Solution Since for any two distinct elements x1 and x2 in 0, 2
π
, sin x1 ≠ sin x2 and
cos x1 ≠ cos x2, both f and g must be one-one But (f + g) (0) = sin 0 + cos 0 = 1 and
(f + g)
π2
= sin
cos
1
2
2
π
π
+
= Therefore, f + g is not one-one Miscellaneous Exercise on Chapter 1
1 |
1 | 719-722 | But (f + g) (0) = sin 0 + cos 0 = 1 and
(f + g)
π2
= sin
cos
1
2
2
π
π
+
= Therefore, f + g is not one-one Miscellaneous Exercise on Chapter 1
1 Show that the function f : R → {x ∈ R : – 1 < x < 1} defined by
( )
1 |
|
x
f x
= +x
,
x ∈ R is one one and onto function |
1 | 720-723 | Therefore, f + g is not one-one Miscellaneous Exercise on Chapter 1
1 Show that the function f : R → {x ∈ R : – 1 < x < 1} defined by
( )
1 |
|
x
f x
= +x
,
x ∈ R is one one and onto function 2 |
1 | 721-724 | Miscellaneous Exercise on Chapter 1
1 Show that the function f : R → {x ∈ R : – 1 < x < 1} defined by
( )
1 |
|
x
f x
= +x
,
x ∈ R is one one and onto function 2 Show that the function f : R → R given by f (x) = x3 is injective |
1 | 722-725 | Show that the function f : R → {x ∈ R : – 1 < x < 1} defined by
( )
1 |
|
x
f x
= +x
,
x ∈ R is one one and onto function 2 Show that the function f : R → R given by f (x) = x3 is injective 3 |
1 | 723-726 | 2 Show that the function f : R → R given by f (x) = x3 is injective 3 Given a non empty set X, consider P(X) which is the set of all subsets of X |
1 | 724-727 | Show that the function f : R → R given by f (x) = x3 is injective 3 Given a non empty set X, consider P(X) which is the set of all subsets of X Define the relation R in P(X) as follows:
For subsets A, B in P(X), ARB if and only if A ⊂ B |
1 | 725-728 | 3 Given a non empty set X, consider P(X) which is the set of all subsets of X Define the relation R in P(X) as follows:
For subsets A, B in P(X), ARB if and only if A ⊂ B Is R an equivalence relation
on P(X) |
1 | 726-729 | Given a non empty set X, consider P(X) which is the set of all subsets of X Define the relation R in P(X) as follows:
For subsets A, B in P(X), ARB if and only if A ⊂ B Is R an equivalence relation
on P(X) Justify your answer |
1 | 727-730 | Define the relation R in P(X) as follows:
For subsets A, B in P(X), ARB if and only if A ⊂ B Is R an equivalence relation
on P(X) Justify your answer 4 |
1 | 728-731 | Is R an equivalence relation
on P(X) Justify your answer 4 Find the number of all onto functions from the set {1, 2, 3, |
1 | 729-732 | Justify your answer 4 Find the number of all onto functions from the set {1, 2, 3, , n} to itself |
1 | 730-733 | 4 Find the number of all onto functions from the set {1, 2, 3, , n} to itself 5 |
1 | 731-734 | Find the number of all onto functions from the set {1, 2, 3, , n} to itself 5 Let A = {– 1, 0, 1, 2}, B = {– 4, – 2, 0, 2} and f, g : A → B be functions defined
by f (x) = x2 – x, x ∈ A and
1
( )
2
1,
2
g x
x
=
−
−
x ∈ A |
1 | 732-735 | , n} to itself 5 Let A = {– 1, 0, 1, 2}, B = {– 4, – 2, 0, 2} and f, g : A → B be functions defined
by f (x) = x2 – x, x ∈ A and
1
( )
2
1,
2
g x
x
=
−
−
x ∈ A Are f and g equal |
1 | 733-736 | 5 Let A = {– 1, 0, 1, 2}, B = {– 4, – 2, 0, 2} and f, g : A → B be functions defined
by f (x) = x2 – x, x ∈ A and
1
( )
2
1,
2
g x
x
=
−
−
x ∈ A Are f and g equal Justify your answer |
1 | 734-737 | Let A = {– 1, 0, 1, 2}, B = {– 4, – 2, 0, 2} and f, g : A → B be functions defined
by f (x) = x2 – x, x ∈ A and
1
( )
2
1,
2
g x
x
=
−
−
x ∈ A Are f and g equal Justify your answer (Hint: One may note that two functions f : A → B and
g : A → B such that f (a) = g(a) ∀ a ∈ A, are called equal functions) |
1 | 735-738 | Are f and g equal Justify your answer (Hint: One may note that two functions f : A → B and
g : A → B such that f (a) = g(a) ∀ a ∈ A, are called equal functions) Rationalised 2023-24
MATHEMATICS
16
6 |
1 | 736-739 | Justify your answer (Hint: One may note that two functions f : A → B and
g : A → B such that f (a) = g(a) ∀ a ∈ A, are called equal functions) Rationalised 2023-24
MATHEMATICS
16
6 Let A = {1, 2, 3} |
1 | 737-740 | (Hint: One may note that two functions f : A → B and
g : A → B such that f (a) = g(a) ∀ a ∈ A, are called equal functions) Rationalised 2023-24
MATHEMATICS
16
6 Let A = {1, 2, 3} Then number of relations containing (1, 2) and (1, 3) which are
reflexive and symmetric but not transitive is
(A) 1
(B) 2
(C) 3
(D) 4
7 |
1 | 738-741 | Rationalised 2023-24
MATHEMATICS
16
6 Let A = {1, 2, 3} Then number of relations containing (1, 2) and (1, 3) which are
reflexive and symmetric but not transitive is
(A) 1
(B) 2
(C) 3
(D) 4
7 Let A = {1, 2, 3} |
1 | 739-742 | Let A = {1, 2, 3} Then number of relations containing (1, 2) and (1, 3) which are
reflexive and symmetric but not transitive is
(A) 1
(B) 2
(C) 3
(D) 4
7 Let A = {1, 2, 3} Then number of equivalence relations containing (1, 2) is
(A) 1
(B) 2
(C) 3
(D) 4
Summary
In this chapter, we studied different types of relations and equivalence relation,
composition of functions, invertible functions and binary operations |
1 | 740-743 | Then number of relations containing (1, 2) and (1, 3) which are
reflexive and symmetric but not transitive is
(A) 1
(B) 2
(C) 3
(D) 4
7 Let A = {1, 2, 3} Then number of equivalence relations containing (1, 2) is
(A) 1
(B) 2
(C) 3
(D) 4
Summary
In this chapter, we studied different types of relations and equivalence relation,
composition of functions, invertible functions and binary operations The main features
of this chapter are as follows:
®
Empty relation is the relation R in X given by R = φ ⊂ X × X |
1 | 741-744 | Let A = {1, 2, 3} Then number of equivalence relations containing (1, 2) is
(A) 1
(B) 2
(C) 3
(D) 4
Summary
In this chapter, we studied different types of relations and equivalence relation,
composition of functions, invertible functions and binary operations The main features
of this chapter are as follows:
®
Empty relation is the relation R in X given by R = φ ⊂ X × X ®
Universal relation is the relation R in X given by R = X × X |
1 | 742-745 | Then number of equivalence relations containing (1, 2) is
(A) 1
(B) 2
(C) 3
(D) 4
Summary
In this chapter, we studied different types of relations and equivalence relation,
composition of functions, invertible functions and binary operations The main features
of this chapter are as follows:
®
Empty relation is the relation R in X given by R = φ ⊂ X × X ®
Universal relation is the relation R in X given by R = X × X ®
Reflexive relation R in X is a relation with (a, a) ∈ R ∀ a ∈ X |
1 | 743-746 | The main features
of this chapter are as follows:
®
Empty relation is the relation R in X given by R = φ ⊂ X × X ®
Universal relation is the relation R in X given by R = X × X ®
Reflexive relation R in X is a relation with (a, a) ∈ R ∀ a ∈ X ®
Symmetric relation R in X is a relation satisfying (a, b) ∈ R implies (b, a) ∈ R |
1 | 744-747 | ®
Universal relation is the relation R in X given by R = X × X ®
Reflexive relation R in X is a relation with (a, a) ∈ R ∀ a ∈ X ®
Symmetric relation R in X is a relation satisfying (a, b) ∈ R implies (b, a) ∈ R ®
Transitive relation R in X is a relation satisfying (a, b) ∈ R and (b, c) ∈ R
implies that (a, c) ∈ R |
1 | 745-748 | ®
Reflexive relation R in X is a relation with (a, a) ∈ R ∀ a ∈ X ®
Symmetric relation R in X is a relation satisfying (a, b) ∈ R implies (b, a) ∈ R ®
Transitive relation R in X is a relation satisfying (a, b) ∈ R and (b, c) ∈ R
implies that (a, c) ∈ R ®
Equivalence relation R in X is a relation which is reflexive, symmetric and
transitive |
1 | 746-749 | ®
Symmetric relation R in X is a relation satisfying (a, b) ∈ R implies (b, a) ∈ R ®
Transitive relation R in X is a relation satisfying (a, b) ∈ R and (b, c) ∈ R
implies that (a, c) ∈ R ®
Equivalence relation R in X is a relation which is reflexive, symmetric and
transitive ®
Equivalence class [a] containing a ∈ X for an equivalence relation R in X is
the subset of X containing all elements b related to a |
1 | 747-750 | ®
Transitive relation R in X is a relation satisfying (a, b) ∈ R and (b, c) ∈ R
implies that (a, c) ∈ R ®
Equivalence relation R in X is a relation which is reflexive, symmetric and
transitive ®
Equivalence class [a] containing a ∈ X for an equivalence relation R in X is
the subset of X containing all elements b related to a ®
A function f : X → Y is one-one (or injective) if
f (x1) = f(x2) ⇒ x1 = x2 ∀ x1, x2 ∈ X |
1 | 748-751 | ®
Equivalence relation R in X is a relation which is reflexive, symmetric and
transitive ®
Equivalence class [a] containing a ∈ X for an equivalence relation R in X is
the subset of X containing all elements b related to a ®
A function f : X → Y is one-one (or injective) if
f (x1) = f(x2) ⇒ x1 = x2 ∀ x1, x2 ∈ X ®
A function f : X → Y is onto (or surjective) if given any y ∈ Y, ∃ x ∈ X such
that f (x) = y |
1 | 749-752 | ®
Equivalence class [a] containing a ∈ X for an equivalence relation R in X is
the subset of X containing all elements b related to a ®
A function f : X → Y is one-one (or injective) if
f (x1) = f(x2) ⇒ x1 = x2 ∀ x1, x2 ∈ X ®
A function f : X → Y is onto (or surjective) if given any y ∈ Y, ∃ x ∈ X such
that f (x) = y ®
A function f : X → Y is one-one and onto (or bijective), if f is both one-one
and onto |
1 | 750-753 | ®
A function f : X → Y is one-one (or injective) if
f (x1) = f(x2) ⇒ x1 = x2 ∀ x1, x2 ∈ X ®
A function f : X → Y is onto (or surjective) if given any y ∈ Y, ∃ x ∈ X such
that f (x) = y ®
A function f : X → Y is one-one and onto (or bijective), if f is both one-one
and onto ®
Given a finite set X, a function f : X → X is one-one (respectively onto) if and
only if f is onto (respectively one-one) |
1 | 751-754 | ®
A function f : X → Y is onto (or surjective) if given any y ∈ Y, ∃ x ∈ X such
that f (x) = y ®
A function f : X → Y is one-one and onto (or bijective), if f is both one-one
and onto ®
Given a finite set X, a function f : X → X is one-one (respectively onto) if and
only if f is onto (respectively one-one) This is the characteristic property of a
finite set |
1 | 752-755 | ®
A function f : X → Y is one-one and onto (or bijective), if f is both one-one
and onto ®
Given a finite set X, a function f : X → X is one-one (respectively onto) if and
only if f is onto (respectively one-one) This is the characteristic property of a
finite set This is not true for infinite set
Rationalised 2023-24
RELATIONS AND FUNCTIONS
17
—v
v
v
v
v—
Historical Note
The concept of function has evolved over a long period of time starting from
R |
1 | 753-756 | ®
Given a finite set X, a function f : X → X is one-one (respectively onto) if and
only if f is onto (respectively one-one) This is the characteristic property of a
finite set This is not true for infinite set
Rationalised 2023-24
RELATIONS AND FUNCTIONS
17
—v
v
v
v
v—
Historical Note
The concept of function has evolved over a long period of time starting from
R Descartes (1596-1650), who used the word ‘function’ in his manuscript
“Geometrie” in 1637 to mean some positive integral power xn of a variable x
while studying geometrical curves like hyperbola, parabola and ellipse |
1 | 754-757 | This is the characteristic property of a
finite set This is not true for infinite set
Rationalised 2023-24
RELATIONS AND FUNCTIONS
17
—v
v
v
v
v—
Historical Note
The concept of function has evolved over a long period of time starting from
R Descartes (1596-1650), who used the word ‘function’ in his manuscript
“Geometrie” in 1637 to mean some positive integral power xn of a variable x
while studying geometrical curves like hyperbola, parabola and ellipse James
Gregory (1636-1675) in his work “ Vera Circuli et Hyperbolae Quadratura”
(1667) considered function as a quantity obtained from other quantities by
successive use of algebraic operations or by any other operations |
1 | 755-758 | This is not true for infinite set
Rationalised 2023-24
RELATIONS AND FUNCTIONS
17
—v
v
v
v
v—
Historical Note
The concept of function has evolved over a long period of time starting from
R Descartes (1596-1650), who used the word ‘function’ in his manuscript
“Geometrie” in 1637 to mean some positive integral power xn of a variable x
while studying geometrical curves like hyperbola, parabola and ellipse James
Gregory (1636-1675) in his work “ Vera Circuli et Hyperbolae Quadratura”
(1667) considered function as a quantity obtained from other quantities by
successive use of algebraic operations or by any other operations Later G |
1 | 756-759 | Descartes (1596-1650), who used the word ‘function’ in his manuscript
“Geometrie” in 1637 to mean some positive integral power xn of a variable x
while studying geometrical curves like hyperbola, parabola and ellipse James
Gregory (1636-1675) in his work “ Vera Circuli et Hyperbolae Quadratura”
(1667) considered function as a quantity obtained from other quantities by
successive use of algebraic operations or by any other operations Later G W |
1 | 757-760 | James
Gregory (1636-1675) in his work “ Vera Circuli et Hyperbolae Quadratura”
(1667) considered function as a quantity obtained from other quantities by
successive use of algebraic operations or by any other operations Later G W Leibnitz (1646-1716) in his manuscript “Methodus tangentium inversa, seu de
functionibus” written in 1673 used the word ‘function’ to mean a quantity varying
from point to point on a curve such as the coordinates of a point on the curve, the
slope of the curve, the tangent and the normal to the curve at a point |
1 | 758-761 | Later G W Leibnitz (1646-1716) in his manuscript “Methodus tangentium inversa, seu de
functionibus” written in 1673 used the word ‘function’ to mean a quantity varying
from point to point on a curve such as the coordinates of a point on the curve, the
slope of the curve, the tangent and the normal to the curve at a point However,
in his manuscript “Historia” (1714), Leibnitz used the word ‘function’ to mean
quantities that depend on a variable |
1 | 759-762 | W Leibnitz (1646-1716) in his manuscript “Methodus tangentium inversa, seu de
functionibus” written in 1673 used the word ‘function’ to mean a quantity varying
from point to point on a curve such as the coordinates of a point on the curve, the
slope of the curve, the tangent and the normal to the curve at a point However,
in his manuscript “Historia” (1714), Leibnitz used the word ‘function’ to mean
quantities that depend on a variable He was the first to use the phrase ‘function
of x’ |
1 | 760-763 | Leibnitz (1646-1716) in his manuscript “Methodus tangentium inversa, seu de
functionibus” written in 1673 used the word ‘function’ to mean a quantity varying
from point to point on a curve such as the coordinates of a point on the curve, the
slope of the curve, the tangent and the normal to the curve at a point However,
in his manuscript “Historia” (1714), Leibnitz used the word ‘function’ to mean
quantities that depend on a variable He was the first to use the phrase ‘function
of x’ John Bernoulli (1667-1748) used the notation φx for the first time in 1718 to
indicate a function of x |
1 | 761-764 | However,
in his manuscript “Historia” (1714), Leibnitz used the word ‘function’ to mean
quantities that depend on a variable He was the first to use the phrase ‘function
of x’ John Bernoulli (1667-1748) used the notation φx for the first time in 1718 to
indicate a function of x But the general adoption of symbols like f, F, φ, ψ |
1 | 762-765 | He was the first to use the phrase ‘function
of x’ John Bernoulli (1667-1748) used the notation φx for the first time in 1718 to
indicate a function of x But the general adoption of symbols like f, F, φ, ψ to
represent functions was made by Leonhard Euler (1707-1783) in 1734 in the first
part of his manuscript “Analysis Infinitorium” |
1 | 763-766 | John Bernoulli (1667-1748) used the notation φx for the first time in 1718 to
indicate a function of x But the general adoption of symbols like f, F, φ, ψ to
represent functions was made by Leonhard Euler (1707-1783) in 1734 in the first
part of his manuscript “Analysis Infinitorium” Later on, Joeph Louis Lagrange
(1736-1813) published his manuscripts “Theorie des functions analytiques” in
1793, where he discussed about analytic function and used the notion f (x), F(x),
φ(x) etc |
1 | 764-767 | But the general adoption of symbols like f, F, φ, ψ to
represent functions was made by Leonhard Euler (1707-1783) in 1734 in the first
part of his manuscript “Analysis Infinitorium” Later on, Joeph Louis Lagrange
(1736-1813) published his manuscripts “Theorie des functions analytiques” in
1793, where he discussed about analytic function and used the notion f (x), F(x),
φ(x) etc for different function of x |
1 | 765-768 | to
represent functions was made by Leonhard Euler (1707-1783) in 1734 in the first
part of his manuscript “Analysis Infinitorium” Later on, Joeph Louis Lagrange
(1736-1813) published his manuscripts “Theorie des functions analytiques” in
1793, where he discussed about analytic function and used the notion f (x), F(x),
φ(x) etc for different function of x Subsequently, Lejeunne Dirichlet
(1805-1859) gave the definition of function which was being used till the set
theoretic definition of function presently used, was given after set theory was
developed by Georg Cantor (1845-1918) |
1 | 766-769 | Later on, Joeph Louis Lagrange
(1736-1813) published his manuscripts “Theorie des functions analytiques” in
1793, where he discussed about analytic function and used the notion f (x), F(x),
φ(x) etc for different function of x Subsequently, Lejeunne Dirichlet
(1805-1859) gave the definition of function which was being used till the set
theoretic definition of function presently used, was given after set theory was
developed by Georg Cantor (1845-1918) The set theoretic definition of function
known to us presently is simply an abstraction of the definition given by Dirichlet
in a rigorous manner |
1 | 767-770 | for different function of x Subsequently, Lejeunne Dirichlet
(1805-1859) gave the definition of function which was being used till the set
theoretic definition of function presently used, was given after set theory was
developed by Georg Cantor (1845-1918) The set theoretic definition of function
known to us presently is simply an abstraction of the definition given by Dirichlet
in a rigorous manner Rationalised 2023-24
18
MATHEMATICS
vMathematics, in general, is fundamentally the science of
self-evident things |
1 | 768-771 | Subsequently, Lejeunne Dirichlet
(1805-1859) gave the definition of function which was being used till the set
theoretic definition of function presently used, was given after set theory was
developed by Georg Cantor (1845-1918) The set theoretic definition of function
known to us presently is simply an abstraction of the definition given by Dirichlet
in a rigorous manner Rationalised 2023-24
18
MATHEMATICS
vMathematics, in general, is fundamentally the science of
self-evident things — FELIX KLEIN v
2 |
1 | 769-772 | The set theoretic definition of function
known to us presently is simply an abstraction of the definition given by Dirichlet
in a rigorous manner Rationalised 2023-24
18
MATHEMATICS
vMathematics, in general, is fundamentally the science of
self-evident things — FELIX KLEIN v
2 1 Introduction
In Chapter 1, we have studied that the inverse of a function
f, denoted by f –1, exists if f is one-one and onto |
1 | 770-773 | Rationalised 2023-24
18
MATHEMATICS
vMathematics, in general, is fundamentally the science of
self-evident things — FELIX KLEIN v
2 1 Introduction
In Chapter 1, we have studied that the inverse of a function
f, denoted by f –1, exists if f is one-one and onto There are
many functions which are not one-one, onto or both and
hence we can not talk of their inverses |
1 | 771-774 | — FELIX KLEIN v
2 1 Introduction
In Chapter 1, we have studied that the inverse of a function
f, denoted by f –1, exists if f is one-one and onto There are
many functions which are not one-one, onto or both and
hence we can not talk of their inverses In Class XI, we
studied that trigonometric functions are not one-one and
onto over their natural domains and ranges and hence their
inverses do not exist |
1 | 772-775 | 1 Introduction
In Chapter 1, we have studied that the inverse of a function
f, denoted by f –1, exists if f is one-one and onto There are
many functions which are not one-one, onto or both and
hence we can not talk of their inverses In Class XI, we
studied that trigonometric functions are not one-one and
onto over their natural domains and ranges and hence their
inverses do not exist In this chapter, we shall study about
the restrictions on domains and ranges of trigonometric
functions which ensure the existence of their inverses and
observe their behaviour through graphical representations |
1 | 773-776 | There are
many functions which are not one-one, onto or both and
hence we can not talk of their inverses In Class XI, we
studied that trigonometric functions are not one-one and
onto over their natural domains and ranges and hence their
inverses do not exist In this chapter, we shall study about
the restrictions on domains and ranges of trigonometric
functions which ensure the existence of their inverses and
observe their behaviour through graphical representations Besides, some elementary properties will also be discussed |
1 | 774-777 | In Class XI, we
studied that trigonometric functions are not one-one and
onto over their natural domains and ranges and hence their
inverses do not exist In this chapter, we shall study about
the restrictions on domains and ranges of trigonometric
functions which ensure the existence of their inverses and
observe their behaviour through graphical representations Besides, some elementary properties will also be discussed The inverse trigonometric functions play an important
role in calculus for they serve to define many integrals |
1 | 775-778 | In this chapter, we shall study about
the restrictions on domains and ranges of trigonometric
functions which ensure the existence of their inverses and
observe their behaviour through graphical representations Besides, some elementary properties will also be discussed The inverse trigonometric functions play an important
role in calculus for they serve to define many integrals The concepts of inverse trigonometric functions is also used in science and engineering |
1 | 776-779 | Besides, some elementary properties will also be discussed The inverse trigonometric functions play an important
role in calculus for they serve to define many integrals The concepts of inverse trigonometric functions is also used in science and engineering 2 |
1 | 777-780 | The inverse trigonometric functions play an important
role in calculus for they serve to define many integrals The concepts of inverse trigonometric functions is also used in science and engineering 2 2 Basic Concepts
In Class XI, we have studied trigonometric functions, which are defined as follows:
sine function, i |
1 | 778-781 | The concepts of inverse trigonometric functions is also used in science and engineering 2 2 Basic Concepts
In Class XI, we have studied trigonometric functions, which are defined as follows:
sine function, i e |
1 | 779-782 | 2 2 Basic Concepts
In Class XI, we have studied trigonometric functions, which are defined as follows:
sine function, i e , sine : R → [– 1, 1]
cosine function, i |
1 | 780-783 | 2 Basic Concepts
In Class XI, we have studied trigonometric functions, which are defined as follows:
sine function, i e , sine : R → [– 1, 1]
cosine function, i e |
1 | 781-784 | e , sine : R → [– 1, 1]
cosine function, i e , cos : R → [– 1, 1]
tangent function, i |
1 | 782-785 | , sine : R → [– 1, 1]
cosine function, i e , cos : R → [– 1, 1]
tangent function, i e |
1 | 783-786 | e , cos : R → [– 1, 1]
tangent function, i e , tan : R – { x : x = (2n + 1) 2
π , n ∈ Z} → R
cotangent function, i |
1 | 784-787 | , cos : R → [– 1, 1]
tangent function, i e , tan : R – { x : x = (2n + 1) 2
π , n ∈ Z} → R
cotangent function, i e |
1 | 785-788 | e , tan : R – { x : x = (2n + 1) 2
π , n ∈ Z} → R
cotangent function, i e , cot : R – { x : x = nπ, n ∈ Z} → R
secant function, i |
1 | 786-789 | , tan : R – { x : x = (2n + 1) 2
π , n ∈ Z} → R
cotangent function, i e , cot : R – { x : x = nπ, n ∈ Z} → R
secant function, i e |
1 | 787-790 | e , cot : R – { x : x = nπ, n ∈ Z} → R
secant function, i e , sec : R – { x : x = (2n + 1) 2
π , n ∈ Z} → R – (– 1, 1)
cosecant function, i |
1 | 788-791 | , cot : R – { x : x = nπ, n ∈ Z} → R
secant function, i e , sec : R – { x : x = (2n + 1) 2
π , n ∈ Z} → R – (– 1, 1)
cosecant function, i e |
1 | 789-792 | e , sec : R – { x : x = (2n + 1) 2
π , n ∈ Z} → R – (– 1, 1)
cosecant function, i e , cosec : R – { x : x = nπ, n ∈ Z} → R – (– 1, 1)
Chapter 2
INVERSE TRIGONOMETRIC
FUNCTIONS
Aryabhata
(476-550 A |
1 | 790-793 | , sec : R – { x : x = (2n + 1) 2
π , n ∈ Z} → R – (– 1, 1)
cosecant function, i e , cosec : R – { x : x = nπ, n ∈ Z} → R – (– 1, 1)
Chapter 2
INVERSE TRIGONOMETRIC
FUNCTIONS
Aryabhata
(476-550 A D |
1 | 791-794 | e , cosec : R – { x : x = nπ, n ∈ Z} → R – (– 1, 1)
Chapter 2
INVERSE TRIGONOMETRIC
FUNCTIONS
Aryabhata
(476-550 A D )
Rationalised 2023-24
INVERSE TRIGONOMETRIC FUNCTIONS 19
We have also learnt in Chapter 1 that if f : X→Y such that f (x) = y is one-one and
onto, then we can define a unique function g : Y→X such that g(y) = x, where x ∈ X
and y = f (x), y ∈ Y |
1 | 792-795 | , cosec : R – { x : x = nπ, n ∈ Z} → R – (– 1, 1)
Chapter 2
INVERSE TRIGONOMETRIC
FUNCTIONS
Aryabhata
(476-550 A D )
Rationalised 2023-24
INVERSE TRIGONOMETRIC FUNCTIONS 19
We have also learnt in Chapter 1 that if f : X→Y such that f (x) = y is one-one and
onto, then we can define a unique function g : Y→X such that g(y) = x, where x ∈ X
and y = f (x), y ∈ Y Here, the domain of g = range of f and the range of g = domain
of f |
1 | 793-796 | D )
Rationalised 2023-24
INVERSE TRIGONOMETRIC FUNCTIONS 19
We have also learnt in Chapter 1 that if f : X→Y such that f (x) = y is one-one and
onto, then we can define a unique function g : Y→X such that g(y) = x, where x ∈ X
and y = f (x), y ∈ Y Here, the domain of g = range of f and the range of g = domain
of f The function g is called the inverse of f and is denoted by f –1 |
1 | 794-797 | )
Rationalised 2023-24
INVERSE TRIGONOMETRIC FUNCTIONS 19
We have also learnt in Chapter 1 that if f : X→Y such that f (x) = y is one-one and
onto, then we can define a unique function g : Y→X such that g(y) = x, where x ∈ X
and y = f (x), y ∈ Y Here, the domain of g = range of f and the range of g = domain
of f The function g is called the inverse of f and is denoted by f –1 Further, g is also
one-one and onto and inverse of g is f |
1 | 795-798 | Here, the domain of g = range of f and the range of g = domain
of f The function g is called the inverse of f and is denoted by f –1 Further, g is also
one-one and onto and inverse of g is f Thus, g–1 = (f –1)–1 = f |
1 | 796-799 | The function g is called the inverse of f and is denoted by f –1 Further, g is also
one-one and onto and inverse of g is f Thus, g–1 = (f –1)–1 = f We also have
(f –1 o f ) (x) = f –1 (f (x)) = f –1(y) = x
and
(f o f –1) (y) = f (f –1(y)) = f (x) = y
Since the domain of sine function is the set of all real numbers and range is the
closed interval [–1, 1] |
1 | 797-800 | Further, g is also
one-one and onto and inverse of g is f Thus, g–1 = (f –1)–1 = f We also have
(f –1 o f ) (x) = f –1 (f (x)) = f –1(y) = x
and
(f o f –1) (y) = f (f –1(y)) = f (x) = y
Since the domain of sine function is the set of all real numbers and range is the
closed interval [–1, 1] If we restrict its domain to
2,
2
−π π
, then it becomes one-one
and onto with range [– 1, 1] |
1 | 798-801 | Thus, g–1 = (f –1)–1 = f We also have
(f –1 o f ) (x) = f –1 (f (x)) = f –1(y) = x
and
(f o f –1) (y) = f (f –1(y)) = f (x) = y
Since the domain of sine function is the set of all real numbers and range is the
closed interval [–1, 1] If we restrict its domain to
2,
2
−π π
, then it becomes one-one
and onto with range [– 1, 1] Actually, sine function restricted to any of the intervals
−
23
2
π
, �π
,
2,
2
−π π
,
2,3
2
π
π
etc |
1 | 799-802 | We also have
(f –1 o f ) (x) = f –1 (f (x)) = f –1(y) = x
and
(f o f –1) (y) = f (f –1(y)) = f (x) = y
Since the domain of sine function is the set of all real numbers and range is the
closed interval [–1, 1] If we restrict its domain to
2,
2
−π π
, then it becomes one-one
and onto with range [– 1, 1] Actually, sine function restricted to any of the intervals
−
23
2
π
, �π
,
2,
2
−π π
,
2,3
2
π
π
etc , is one-one and its range is [–1, 1] |
1 | 800-803 | If we restrict its domain to
2,
2
−π π
, then it becomes one-one
and onto with range [– 1, 1] Actually, sine function restricted to any of the intervals
−
23
2
π
, �π
,
2,
2
−π π
,
2,3
2
π
π
etc , is one-one and its range is [–1, 1] We can,
therefore, define the inverse of sine function in each of these intervals |
1 | 801-804 | Actually, sine function restricted to any of the intervals
−
23
2
π
, �π
,
2,
2
−π π
,
2,3
2
π
π
etc , is one-one and its range is [–1, 1] We can,
therefore, define the inverse of sine function in each of these intervals We denote the
inverse of sine function by sin–1 (arc sine function) |
1 | 802-805 | , is one-one and its range is [–1, 1] We can,
therefore, define the inverse of sine function in each of these intervals We denote the
inverse of sine function by sin–1 (arc sine function) Thus, sin–1 is a function whose
domain is [– 1, 1] and range could be any of the intervals
23 ,
2
− π −π
,
2,
2
−π π
or
2,3
2
π
π
, and so on |
1 | 803-806 | We can,
therefore, define the inverse of sine function in each of these intervals We denote the
inverse of sine function by sin–1 (arc sine function) Thus, sin–1 is a function whose
domain is [– 1, 1] and range could be any of the intervals
23 ,
2
− π −π
,
2,
2
−π π
or
2,3
2
π
π
, and so on Corresponding to each such interval, we get a branch of the
function sin–1 |
1 | 804-807 | We denote the
inverse of sine function by sin–1 (arc sine function) Thus, sin–1 is a function whose
domain is [– 1, 1] and range could be any of the intervals
23 ,
2
− π −π
,
2,
2
−π π
or
2,3
2
π
π
, and so on Corresponding to each such interval, we get a branch of the
function sin–1 The branch with range
2,
2
−π π
is called the principal value branch,
whereas other intervals as range give different branches of sin–1 |
1 | 805-808 | Thus, sin–1 is a function whose
domain is [– 1, 1] and range could be any of the intervals
23 ,
2
− π −π
,
2,
2
−π π
or
2,3
2
π
π
, and so on Corresponding to each such interval, we get a branch of the
function sin–1 The branch with range
2,
2
−π π
is called the principal value branch,
whereas other intervals as range give different branches of sin–1 When we refer
to the function sin–1, we take it as the function whose domain is [–1, 1] and range is
2,
2
−π π
|
1 | 806-809 | Corresponding to each such interval, we get a branch of the
function sin–1 The branch with range
2,
2
−π π
is called the principal value branch,
whereas other intervals as range give different branches of sin–1 When we refer
to the function sin–1, we take it as the function whose domain is [–1, 1] and range is
2,
2
−π π
We write sin–1 : [–1, 1] →
2,
2
−π π
From the definition of the inverse functions, it follows that sin (sin–1 x) = x
if – 1 ≤ x ≤ 1 and sin–1 (sin x) = x if
2
2
x
π
π
−
≤
≤ |
1 | 807-810 | The branch with range
2,
2
−π π
is called the principal value branch,
whereas other intervals as range give different branches of sin–1 When we refer
to the function sin–1, we take it as the function whose domain is [–1, 1] and range is
2,
2
−π π
We write sin–1 : [–1, 1] →
2,
2
−π π
From the definition of the inverse functions, it follows that sin (sin–1 x) = x
if – 1 ≤ x ≤ 1 and sin–1 (sin x) = x if
2
2
x
π
π
−
≤
≤ In other words, if y = sin–1 x, then
sin y = x |
1 | 808-811 | When we refer
to the function sin–1, we take it as the function whose domain is [–1, 1] and range is
2,
2
−π π
We write sin–1 : [–1, 1] →
2,
2
−π π
From the definition of the inverse functions, it follows that sin (sin–1 x) = x
if – 1 ≤ x ≤ 1 and sin–1 (sin x) = x if
2
2
x
π
π
−
≤
≤ In other words, if y = sin–1 x, then
sin y = x Remarks
(i)
We know from Chapter 1, that if y = f(x) is an invertible function, then x = f –1 (y) |
1 | 809-812 | We write sin–1 : [–1, 1] →
2,
2
−π π
From the definition of the inverse functions, it follows that sin (sin–1 x) = x
if – 1 ≤ x ≤ 1 and sin–1 (sin x) = x if
2
2
x
π
π
−
≤
≤ In other words, if y = sin–1 x, then
sin y = x Remarks
(i)
We know from Chapter 1, that if y = f(x) is an invertible function, then x = f –1 (y) Thus, the graph of sin–1 function can be obtained from the graph of original
function by interchanging x and y axes, i |
1 | 810-813 | In other words, if y = sin–1 x, then
sin y = x Remarks
(i)
We know from Chapter 1, that if y = f(x) is an invertible function, then x = f –1 (y) Thus, the graph of sin–1 function can be obtained from the graph of original
function by interchanging x and y axes, i e |
1 | 811-814 | Remarks
(i)
We know from Chapter 1, that if y = f(x) is an invertible function, then x = f –1 (y) Thus, the graph of sin–1 function can be obtained from the graph of original
function by interchanging x and y axes, i e , if (a, b) is a point on the graph of
sine function, then (b, a) becomes the corresponding point on the graph of inverse
Rationalised 2023-24
20
MATHEMATICS
of sine function |
1 | 812-815 | Thus, the graph of sin–1 function can be obtained from the graph of original
function by interchanging x and y axes, i e , if (a, b) is a point on the graph of
sine function, then (b, a) becomes the corresponding point on the graph of inverse
Rationalised 2023-24
20
MATHEMATICS
of sine function Thus, the graph of the function y = sin–1 x can be obtained from
the graph of y = sin x by interchanging x and y axes |
1 | 813-816 | e , if (a, b) is a point on the graph of
sine function, then (b, a) becomes the corresponding point on the graph of inverse
Rationalised 2023-24
20
MATHEMATICS
of sine function Thus, the graph of the function y = sin–1 x can be obtained from
the graph of y = sin x by interchanging x and y axes The graphs of y = sin x and
y = sin–1 x are as given in Fig 2 |
1 | 814-817 | , if (a, b) is a point on the graph of
sine function, then (b, a) becomes the corresponding point on the graph of inverse
Rationalised 2023-24
20
MATHEMATICS
of sine function Thus, the graph of the function y = sin–1 x can be obtained from
the graph of y = sin x by interchanging x and y axes The graphs of y = sin x and
y = sin–1 x are as given in Fig 2 1 (i), (ii), (iii) |
1 | 815-818 | Thus, the graph of the function y = sin–1 x can be obtained from
the graph of y = sin x by interchanging x and y axes The graphs of y = sin x and
y = sin–1 x are as given in Fig 2 1 (i), (ii), (iii) The dark portion of the graph of
y = sin–1 x represent the principal value branch |
1 | 816-819 | The graphs of y = sin x and
y = sin–1 x are as given in Fig 2 1 (i), (ii), (iii) The dark portion of the graph of
y = sin–1 x represent the principal value branch (ii)
It can be shown that the graph of an inverse function can be obtained from the
corresponding graph of original function as a mirror image (i |
1 | 817-820 | 1 (i), (ii), (iii) The dark portion of the graph of
y = sin–1 x represent the principal value branch (ii)
It can be shown that the graph of an inverse function can be obtained from the
corresponding graph of original function as a mirror image (i e |
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