Maths-
General
Easy

Question

If f left parenthesis x right parenthesis equals vertical line x minus 1 vertical line plus vertical line x minus 2 vertical line plus vertical line x minus 3 vertical line comma f colon left square bracket 2 comma 3 right square bracket not stretchy rightwards arrow R text  is  end text

  1. one-one onto function
  2. an onto function only
  3. an identity function
  4. an into function only

hintHint:

In this question, we have to find the type of function if f left parenthesis x right parenthesis equals vertical line x minus 1 vertical line plus vertical line x minus 2 vertical line plus vertical line x minus 3 vertical line comma f colon left square bracket 2 comma 3 right square bracket not stretchy rightwards arrow 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, 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, and a function is said to be an identity function if the function is both one-one and onto.

The correct answer is: an identity function


    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 left parenthesis x minus 1 right parenthesis plus left parenthesis x minus 2 right parenthesis minus left parenthesis x minus 3 right parenthesis comma space i f space 2 less or equal than x less or equal than 3 end cell end table close
C a s e space i comma space w h e n space space space 2 less or equal than x less or equal than 3
f left parenthesis x right parenthesis equals left parenthesis x minus 1 right parenthesis plus left parenthesis x plus 2 right parenthesis minus left parenthesis x minus 3 right parenthesis
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 left parenthesis x subscript 1 minus 1 right parenthesis plus left parenthesis x subscript 1 plus 2 right parenthesis minus left parenthesis x subscript 1 minus 3 right parenthesis equals left parenthesis x subscript 2 minus 1 right parenthesis plus left parenthesis x subscript 2 plus 2 right parenthesis minus left parenthesis x subscript 2 minus 3 right parenthesis
rightwards double arrow x subscript 1 minus 1 plus x subscript 1 plus 2 minus x subscript 1 plus 3 equals x subscript 2 minus 1 plus x subscript 2 plus 2 minus x subscript 2 plus 3
rightwards double arrow x subscript 1 plus 4 equals x subscript 2 plus 4
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 left parenthesis x minus 1 right parenthesis plus left parenthesis x plus 2 right parenthesis minus left parenthesis x minus 3 right parenthesis
rightwards double arrow y equals x minus 1 plus x plus 2 minus x plus 3
rightwards double arrow y equals x plus 4
rightwards double arrow x equals y minus 4
N o w comma space f open parentheses y minus 4 close parentheses equals left parenthesis y minus 4 minus 1 right parenthesis plus left parenthesis y minus 4 plus 2 right parenthesis minus left parenthesis y minus 4 minus 3 right parenthesis equals y minus 5 plus y minus 2 minus y plus 7 equals y
S o comma space f left parenthesis x right parenthesis space i s space o n t o.
S o comma space i t space i s space i d e n t i t y space f u n c t i o.

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