Maths-
General
Easy

Question

y equals f left parenthesis x right parenthesis equals fraction numerator x over denominator 1 plus vertical line x vertical line end fraction comma x element of R comma y element of R text  is  end text

  1. one-one and onto

  2. one-one and onto

  3. one-one but not onto

  4. one one onto

hintHint:

In this question, we have to find the type of function i f space y equals f left parenthesis x right parenthesis equals fraction numerator x over denominator 1 plus vertical line x vertical line end fraction comma x element of R comma y element of R.  A function f colon A rightwards arrow B is said to be one-one if different elements of A have different images in B, and a function f colon A rightwards arrow B is said to be onto if for all y element of B, there exists x element of A such that f (x) = y.

The correct answer is:

one-one and onto


    w e space h a v e comma space
f left parenthesis x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator x over denominator 1 plus x end fraction comma space i f space x greater or equal than 0 end cell row cell fraction numerator x over denominator 1 minus x end fraction comma space i f space x less or equal than 0 end cell end table close
C a s e space i comma space w h e n space x greater or equal than 0
f left parenthesis x right parenthesis equals fraction numerator x over denominator 1 plus x end fraction
t h e n comma space f left parenthesis x subscript 1 right parenthesis equals f left parenthesis x subscript 2 right parenthesis rightwards double arrow fraction numerator x subscript 1 over denominator 1 plus x subscript 1 end fraction equals fraction numerator x subscript 2 over denominator 1 plus x subscript 2 end fraction rightwards double arrow x subscript 1 plus x subscript 1 x subscript 2 equals x subscript 2 plus x subscript 1 x subscript 2 rightwards double arrow x subscript 1 equals x subscript 2
S o comma space f space i s space o n e minus space o n e.
L e t space y element of R
a n d space y equals fraction numerator x over denominator 1 plus x end fraction rightwards double arrow y plus x y equals x rightwards double arrow x left parenthesis 1 minus y right parenthesis equals y rightwards double arrow x equals fraction numerator y over denominator 1 minus y end fraction
N o w comma space f open parentheses fraction numerator y over denominator 1 minus y end fraction close parentheses equals fraction numerator fraction numerator y over denominator 1 minus y end fraction over denominator 1 plus fraction numerator y over denominator 1 minus y end fraction end fraction equals fraction numerator y over denominator 1 minus y plus y end fraction equals y
S o comma space f left parenthesis x right parenthesis space i s space o n t o.

C a s e space i comma space w h e n space x less or equal than 0
f left parenthesis x right parenthesis equals fraction numerator x over denominator 1 minus x end fraction
t h e n comma space f left parenthesis x subscript 1 right parenthesis equals f left parenthesis x subscript 2 right parenthesis rightwards double arrow fraction numerator x subscript 1 over denominator 1 minus x subscript 1 end fraction equals fraction numerator x subscript 2 over denominator 1 minus x subscript 2 end fraction rightwards double arrow x subscript 1 minus x subscript 1 x subscript 2 equals x subscript 2 minus x subscript 1 x subscript 2 rightwards double arrow x subscript 1 equals x subscript 2
S o comma space f space i s space o n e minus space o n e.
L e t space y element of R
a n d space y equals fraction numerator x over denominator 1 minus x end fraction rightwards double arrow y minus x y equals x rightwards double arrow x left parenthesis 1 plus y right parenthesis equals y rightwards double arrow x equals fraction numerator y over denominator 1 plus y end fraction
N o w comma space f open parentheses fraction numerator y over denominator 1 plus y end fraction close parentheses equals fraction numerator fraction numerator y over denominator 1 plus y end fraction over denominator 1 minus fraction numerator y over denominator 1 plus y end fraction end fraction equals fraction numerator y over denominator 1 plus y minus y end fraction equals y
S o comma space f left parenthesis x right parenthesis space i s space o n t o.

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