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Question 4

 If f(x), show that fof(x) = x, for all x ≠ 2/3. What is the inverse of f ?

Answer

It is given that.

Hence, the given function f is invertible and the inverse of f is f itself.

 

 

Popular Questions of Class 12th mathematics

 

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Let A = R – {3} and B = R – {1}. Consider the function  f : A → B defined by

. Is f one-one and onto? Justify your answer. 

 

  • Q:-

     Prove that the Greatest Integer Function f : R → R, given by f(x) = [x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x.

  • Q:- Show that the relation R in the set R of real numbers, defined as R = {(a, b): a ≤ b2} is neither reflexive nor symmetric nor transitive.
  • Q:-

     Show that the Modulus Function f : R → R, given by f(x) = |x|, is neither oneone nor onto, where | x | is x, if x is positive or 0 and |x| is – x, if x is negative.

  • Q:- Show that the relation R in the set A = {1, 2, 3, 4, 5} given by R = { (a,b) ; |a - b| is even}, is an equivalence relation. Show that all the elements of {1, 3, 5} are related to each other and all the elements of {2, 4} are related to each other. But no element of {1, 3, 5} is related to any element of {2, 4}.
  •  

    ">

    Let A = R – {3} and B = R – {1}. Consider the function  f : A → B defined by

    . Is f one-one and onto? Justify your answer. 

     

  • Q:- Determine whether each of the following relations are reflexive, symmetric and transitive:
    (i) Relation R in the set A = {1, 2, 3,13, 14} defined as
    R = {(x, y): 3x − y = 0}
    (ii) Relation R in the set N of natural numbers defined as
    R = {(x, y): y = x + 5 and x < 4}
    (iii) Relation R in the set A = {1, 2, 3, 4, 5, 6} as
    R = {(x, y): y is divisible by x}
    (iv) Relation R in the set Z of all integers defined as
    R = {(x, y): x − y is as integer}
    (v) Relation R in the set A of human beings in a town at a particular time given by
    (a) R = {(x, y): x and y work at the same place}
    (b) R = {(x, y): x and y live in the same locality}
    (c) R = {(x, y): x is exactly 7 cm taller than y}
    (d) R = {(x, y): x is wife of y}
    (e) R = {(x, y): x is father of y}
  • Q:-

    Show that the function f : R* → R* defined by f(x) = 1/x is one-one and onto,where R* is the set of all non-zero real numbers. Is the result true, if the domain R* is replaced by N with co-domain being same as R

  • Q:-

    Answer the following as true or false.

     \begin{align}(i) \overrightarrow{a}\;  and\; \overrightarrow{-a}\; are\; collinear.\end{align}

    (ii) Two collinear vectors are always equal in magnitude.

    (iii) Two vectors having same magnitude are collinear.

    (iv) Two collinear vectors having the same magnitude are equal.

  • Q:- Let L be the set of all lines in XY plane and R be the relation in L defined as R = {(L1, L2): L1 is parallel to L2}. Show that R is an equivalence relation. Find the set of all lines related to the line y = 2x + 4.
  • Q:- sin 2x – 4e3x
  • Q:- Show that the relation R in the set A = {1, 2, 3, 4, 5} given by R = { (a,b) ; |a - b| is even}, is an equivalence relation. Show that all the elements of {1, 3, 5} are related to each other and all the elements of {2, 4} are related to each other. But no element of {1, 3, 5} is related to any element of {2, 4}.
  • Q:-

    \begin{align} y = xsinx:xy{'}=y +x\sqrt{x^2 -y^2}(x\neq0\; and\; x>y\; or\; x<-y)\end{align}

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