Maths-
General
Easy

Question

If f colon R not stretchy rightwards arrow left parenthesis 0 comma 1 right square bracket defined by f left parenthesis x right parenthesis equals fraction numerator 1 over denominator x squared plus 1 end fraction comma text  then  end text f text  is  end text

  1. not one-one
  2. not onto
  3. not one-one but onto
  4. one-one but not onto

hintHint:

In this question, we have to find the type of function if f colon R not stretchy rightwards arrow left parenthesis 0 comma 1 right square bracket defined by f left parenthesis x right parenthesis equals fraction numerator 1 over denominator x squared plus 1 end fraction. 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: not one-one but onto


    f colon R rightwards arrow left parenthesis 0 comma 1 right square bracket space comma space f space left parenthesis x right parenthesis equals fraction numerator 1 over denominator x squared plus 1 end fraction
w e space h a v e comma space f left parenthesis x subscript 1 right parenthesis space equals f left parenthesis x subscript 2 right parenthesis
rightwards double arrow fraction numerator 1 over denominator x subscript 1 squared plus 1 end fraction equals fraction numerator 1 over denominator x subscript 2 squared plus 1 end fraction
rightwards double arrow x subscript 2 squared plus 1 equals x subscript 1 squared plus 1
rightwards double arrow x subscript 2 squared equals x subscript 1 squared
rightwards double arrow x subscript 1 squared minus x subscript 2 squared equals 0
rightwards double arrow left parenthesis x subscript 1 minus x subscript 2 right parenthesis plus left parenthesis x subscript 1 plus x subscript 2 right parenthesis equals 0
rightwards double arrow x subscript 1 equals x subscript 2 space o r space x subscript 1 equals space minus x subscript 2
s o comma space f space i s space n o t space o n e minus o n e.
L e t space y element of R rightwards arrow left parenthesis 0 comma 1 right square bracket space
a n d space y equals fraction numerator 1 over denominator x squared plus 1 end fraction comma
rightwards double arrow y x squared plus y equals 1
rightwards double arrow x equals square root of fraction numerator 1 minus y over denominator y end fraction end root
N o w comma space f open parentheses square root of fraction numerator 1 minus y over denominator y end fraction end root close parentheses space equals fraction numerator 1 over denominator open parentheses square root of fraction numerator 1 minus y over denominator y end fraction end root close parentheses squared plus 1 end fraction equals fraction numerator 1 over denominator begin display style fraction numerator 1 minus y over denominator y end fraction end style plus 1 end fraction equals fraction numerator y over denominator 1 minus y plus y end fraction equals y
S o comma space f space i s space o n t o space f u n c t i o n. space

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