(iv) Composition (Function of Function) Let f : A → B and g : B → C be two functions. g y y
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Let A and B be .two non-empty sets, then a function f from set A to set B is a rule whichassociates each element of A to a unique element of B. ... Download NCERT Solutions for Class 12 Chapter 1 Relations and Functions. We define g : B → A, such that f(x) = y => g(y) = x, g is called inverse of f and vice-versa. R = {(P, Q) : distance of the point P from the origin is same as the distance of the
(D) R is an equivalence relation. (i) Put f(T + x) = f(x) and solve this equation to find the positive values of T independent of x. (b) R = {(x, y) : x and y live in the same locality}
NCERT Solutions for Class 12 Maths Chapter-1 PDF is now available for download on the official website of Vedantu. %PDF-1.4 Question 10. . Question 5. 0000008434 00000 n
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(ii) Implicit Functions A function is said to be an implicit function, if it is expressed in the
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(B) Is commutative but not associative? (ix) [x + y] = [x] + [y + x – [x}] for all x, y ∈ R
Range of f(x) = R – {0}, Modulus function is given by y = f(x) = |x| , where |x| denotes the absolute value of x, that is, Domain of f(x) = R
(ii) g : {5, 6, 7, 8} → {1, 2, 3, 4} with
(iv) Relation R in the set Z of all integers defined as
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inverse of f. Question 8. Range f(x) = R. If a and b be fixed real numbers, then the linear function is defmed as y = f(x) = ax + b, where a
(ii) On Z+, define by a b = ab
Question 14.Let L be the set of all lines in XY plane and R be the relation in L defined as
(i) Explicit Functions A function is said to be an explicit function, if it is expressed in the form
(A) R is reflexive and symmetric but not transitive. (C) R is symmetric and transitive but not reflexive. 0000005901 00000 n
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Ncert Math Notes For Class 12 Chapter 1 Relations And Functions Download PDf.
(v) Odd function * Odd function = Even function. − ,
(i) Even function ± Even function = Even function. ∴ is reflexive. In case, the domain of f
function, then d / dx f(x) or ∫ f(x) dx is even. related to each other. (x) [x + y] ≥ [x] + [y]
Let f : X → Y be an invertible function. (ix) tann x and cotnx are periodic functions with period π. (iii) Irrational Functions The algebraic functions containing one or more terms having nonintegral
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Odd Functions A real function f(x) is an odd function, if f( -x) = – f(x). (4 3)
Hence, is reflexive and transitive but not symmetric. operation? Question 5.Show that the Signum Function f : R→R, given by
Relations and Functions Class 12 Maths MCQs Pdf. (vii) |sin x|, |cos x|, |tan x|, |cot x|, |sec x| and |cosec x| are periodic functions with period π. endobj
(xi) An even function can never be one-one, however an odd function mayor may not be oneone. h = {(2, 7), (3, 9), (4, 11), (5, 13)}, Question 6. Question 9. Show that the relation R in the set A of all the books in a library of a college,
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∴R is not reflexive. (vi) gof or fog is even, if anyone of f and g or both are even. (iii) Compute (2 3) (4 5). Let f : N → N be defined by f (n) =
Question 1.Let f : {1, 3, 4} → {1, 2, 5} and g : {1, 2, 5} → {1, 3} be given by
Question 11. Let R be the relation in the set N given by R = {(a, b) : a = b – 2, b > 6}. –1, if 0
Range of f(x) = (0, ∞), Domain of f(x) = R – {0}
(ii) Transitive but neither reflexive nor symmetric. 0000013697 00000 n
and semi-closed intervals. Is the operation ′ same as the operation defined
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Show that none of the operations given above has identity. 1
(iii) Period = 2π, (i) Domain = R ~ (2n + 1) π / 2, n ∈ I
Even Functions A real function f(x) is an even function, if f( -x) = f(x). x
a < 1. Question 3.Prove that the Greatest Integer Function f : R→R, given by f (x) = [x], is neither
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fog1(y) = 1Y(y) = fog2(y). 0000011320 00000 n
Consider f : R+→ [4, ∞) given by f (x) = x2 +4..Show that f is invertible with the
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In Chapter 1, you 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. (C) f is one-one but not onto
which is increasing, if a > 1 and decreasing, if 0 <
(ii) Graphically If any .line parallel to x-axis cuts the graph of the function atmost at one point,
&��:�/�W5���Ktr�9ٳ)?�'�(��j~_&�J�����U��g[��Ka�r��=�u��:���� �Ny�Y`���1ȝ������\2��hY�Ng�z� ��t7 Let R be a relation on the set L of lines defined by l 1 R l 2 if l 1 is perpendicular to l 2, then relation R is (a) reflexive and symmetric (b) symmetric and transitive