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Maths-

General

Easy

Question

Which of the following functions is not injective?

$f (x) = | x + 1 |, x \in [- 1, \infty)$
$g (x) = x + \frac{1}{x}, x \in (0, \infty)$
$h (x) = x^{2} + 4 x - 5, x \in (0, \infty)$
$k (x) = e^{- x}, x \in [0, \infty)$

hint Hint:

An injective function (i.e., one-to-one function) is a function for which every element of the range of the function corresponds to exactly one element of the domain. For finding whether the function is injective, we will first assume $f colon N rightwards arrow N colon f left parenthesis x right parenthesis comma space T h e n comma space f left parenthesis x subscript 1 right parenthesis equals f open parentheses x subscript 2 close parentheses comma space i f space x subscript 1 equals x subscript 2$ then the function is one-one or injective function.

The correct answer is: $g left parenthesis x right parenthesis equals x plus 1 over x comma x element of left parenthesis 0 comma straight infinity right parenthesis$

$F o r comma space f left parenthesis x right parenthesis equals open vertical bar x plus 1 close vertical bar comma space x element of left square bracket negative 1 comma infinity right parenthesis A s space t h e space r a n g e space o f space x space i s space left square bracket negative 1 comma infinity right parenthesis space s o comma space x plus 1 space greater or equal than 0 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 space space x subscript 1 plus 1 equals x subscript 2 plus 1 rightwards double arrow space space space space space space space x subscript 1 equals x subscript 2 S o comma space f left parenthesis x right parenthesis space i s space i n j e c t i v e space f u n c t i o n space. F o r comma space g left parenthesis x right parenthesis equals x plus fraction numerator 1 over denominator open vertical bar x close vertical bar end fraction comma space x element of left parenthesis 0 comma infinity right parenthesis A s space t h e space r a n g e space o f space x space i s space left parenthesis 0 comma infinity right parenthesis space s o comma space x greater than 0 t h e n comma space g left parenthesis x subscript 1 right parenthesis equals g left parenthesis x subscript 2 right parenthesis rightwards double arrow space space x subscript 1 plus 1 over x subscript 1 equals x subscript 2 plus 1 over x subscript 2 rightwards double arrow fraction numerator x subscript 1 squared plus 1 over denominator x subscript 1 end fraction equals fraction numerator x subscript 2 squared plus 1 over denominator x subscript 2 end fraction rightwards double arrow x subscript 1 squared. x subscript 2 plus x subscript 2 equals x subscript 2 squared. x subscript 1 plus x subscript 1 rightwards double arrow x subscript 1 squared. x subscript 2 plus x subscript 2 squared. x subscript 1 plus x subscript 1 plus x subscript 2 equals 0 rightwards double arrow x subscript 1. end subscript x subscript 2 left parenthesis x subscript 1 plus x subscript 2 right parenthesis plus open parentheses x subscript 1 plus x subscript 2 close parentheses equals 0 rightwards double arrow left parenthesis x subscript 1 plus x subscript 2 right parenthesis left parenthesis x subscript 1. x subscript 2 plus 1 right parenthesis equals 0 A s comma space left parenthesis x subscript 1 plus x subscript 2 right parenthesis not equal to 0 comma space left parenthesis x subscript 1. x subscript 2 plus 1 right parenthesis not equal to 0 space t h e n comma space g left parenthesis x subscript 1 right parenthesis equals g left parenthesis x subscript 2 right parenthesis space i s space f a l s e. S o comma space g left parenthesis x right parenthesis space i s space n o t space i n j e c t i v e space f u n c t i o n space. F o r comma space h left parenthesis x right parenthesis equals x squared plus 4 x minus 5 comma space x element of left parenthesis 0 comma infinity right parenthesis A s space t h e space r a n g e space o f space x space i s space left square bracket negative 1 comma infinity right parenthesis space s o comma space x plus 1 space greater or equal than 0 t h e n comma space h left parenthesis x subscript 1 right parenthesis equals h left parenthesis x subscript 2 right parenthesis rightwards double arrow space x subscript 1 squared plus 4 space x subscript 1 minus 5 equals space x subscript 2 squared plus 4 space x subscript 2 minus 5 rightwards double arrow space x subscript 1 squared plus 4 space x subscript 1 equals space x subscript 2 squared plus 4 space x subscript 2 rightwards double arrow left parenthesis x subscript 1 squared minus x subscript 2 squared right parenthesis plus 4 open parentheses x subscript 1 minus x subscript 2 close parentheses equals 0 rightwards double arrow open parentheses x subscript 1 minus x subscript 2 close parentheses open parentheses x subscript 1 minus x subscript 2 plus 1 close parentheses equals 0 a s comma space open parentheses x subscript 1 minus x subscript 2 plus 1 close parentheses space not equal to 0 rightwards double arrow open parentheses x subscript 1 minus x subscript 2 close parentheses equals 0 rightwards double arrow space space space space space space space x subscript 1 equals x subscript 2 S o comma space h left parenthesis x right parenthesis space i s space i n j e c t i v e space f u n c t i o n space. F o r comma space k left parenthesis x right parenthesis equals e to the power of negative x end exponent comma space x element of left square bracket 0 comma infinity right parenthesis t h e n comma space k left parenthesis x subscript 1 right parenthesis equals k left parenthesis x subscript 2 right parenthesis rightwards double arrow space e to the power of negative x subscript 1 end exponent equals e to the power of negative x subscript 2 end exponent rightwards double arrow negative x subscript 1 log space e space equals negative x subscript 2 space log space e rightwards double arrow space minus space x subscript 1 equals negative x subscript 2 rightwards double arrow space space x subscript 1 equals x subscript 2 S o comma space k left parenthesis x right parenthesis space i s space i n j e c t i v e space f u n c t i o n space.$

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