Maths-
General
Easy

Question

The mapping f colon N not stretchy rightwards arrow N given f left parenthesis n right parenthesis equals 1 plus n squared comma n element of N where N is the set of natural number, is

  1. One-to-one and onto
  2. Onto but not one-to-one
  3. One-to-one but not onto
  4. Neither one-to-one nor onto

The correct answer is: One-to-one but not onto


    Since, f left parenthesis n right parenthesis equals 1 plus n squared
    For one-to-one, 1 plus n subscript 1 superscript 2 equals 1 plus n subscript 2 superscript 2
    table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row cell not stretchy rightwards double arrow n subscript 1 superscript 2 minus n subscript 2 superscript 2 equals 0 end cell row cell not stretchy rightwards double arrow n subscript 1 equals n subscript 2 open parentheses because n subscript 1 plus n subscript 2 not equal to 0 close parentheses end cell end table
    therefore f left parenthesis n right parenthesis is one-to-one.
    But f left parenthesis n right parenthesis is not onto as every element of codomain is not the image of any element of domains.
    Hence, f left parenthesis n right parenthesis is one-to-one but not onto

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