Follow Us


Question 6

Show that f : [–1, 1] → R, given by is one-one. Find the inverse of the function f : [–1, 1] → Range f.

(Hint: For y ∈ Range fy =, for some x in [ - 1, 1], i.e.,)

 

Answer

f: [ - 1, 1] → R is given as

Let f(x) = f(y).

∴ f is a one-one function.

It is clear that f: [ - 1, 1] → Range f is onto.

∴ f: [ - 1, 1]→ Range f is one-one and onto and therefore, the inverse of the function:

f: [ - 1, 1] → Range f exists.

Let g: Range f → [ - 1, 1] be the inverse of f.

Let y be an arbitrary element of range f.

Since f: [ - 1, 1] → Range f is onto, we have:

Now, let us define g: Range f → [ - 1, 1] as

gof =I[-1, 1]and fog = IRange f

∴ f - 1 = g

⇒ 

  

 

 

Popular Questions of Class 12th mathematics

 

">

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:- 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:-

     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 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 R of real numbers, defined as R = {(a, b): a ≤ b2} is neither reflexive nor symmetric nor transitive.
  •  

    ">

    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:-

    If a line has the direction ratios −18, 12, −4, then what are its direction cosines?

  • Q:-

     Find gof and fog, if

    (i) f(x) = | x | and g(x) = | 5x – 2 |
    (ii) f(x) = 8x3 and g(x) = x1/3 .

     

  • Q:-

    Let f: X → Y be an invertible function. Show that the inverse of f –1 is f, i.e., (f–1)–1 = f.

  • Q:-

    If a line makes angles 90°, 135°, 45° with x, y and z-axes respectively, find its direction cosines.

     

  • 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:-

    Check the injectivity and surjectivity of the following functions:

    (i) f : N → N given by f(x) = x2

    (ii) f : Z → Z given by f(x) = x2

    (iii) f : R → R given by f(x) = x2

    (iv) f : N → N given by f(x) = x3

    (v) f : Z → Z given by f(x) = x

  • Q:-

    In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.

    (i) f : R → R defined by f(x) = 3 – 4x

    (ii) f : R → R defined by f(x) = 1 + x

  • 1 Comment(s) on this Question

    Write a Comment: