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Maths-

General

Easy

Question

If $f colon R not stretchy rightwards arrow R$ is defined by $f left parenthesis x right parenthesis equals fraction numerator x squared minus 4 over denominator x squared plus 1 end fraction$ $text then end text f left parenthesis x right parenthesis text is end text$

one-one and not onto
one-one and onto
not one-one but onto
neither one-one nor onto

hint Hint:

In this question, we have to find the type of function if $f colon R not stretchy rightwards arrow R$ is defined by $f left parenthesis x right parenthesis equals fraction numerator x squared minus 4 over denominator x squared plus 1 end fraction$ . 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 not stretchy rightwards arrow R comma space f left parenthesis x right parenthesis equals fraction numerator x squared minus 4 over denominator x squared plus 1 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 squared minus 4 over denominator x subscript 1 squared plus 1 end fraction equals fraction numerator x subscript 2 squared minus 4 over denominator x subscript 2 squared plus 1 end fraction rightwards double arrow x subscript 1 squared x subscript 2 squared minus 4 x subscript 2 squared plus x subscript 1 squared minus 4 equals x subscript 1 squared x subscript 2 squared minus 4 x subscript 1 squared plus x subscript 2 squared minus 4 rightwards double arrow 5 x subscript 1 squared space minus 5 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 comma a n d space y equals fraction numerator x squared minus 4 over denominator x squared plus 1 end fraction rightwards double arrow x squared y plus y equals x squared minus 4 rightwards double arrow 4 plus y equals x squared minus x squared y rightwards double arrow x squared equals fraction numerator 4 plus y over denominator 1 minus y end fraction rightwards double arrow x equals square root of fraction numerator 4 plus y over denominator 1 minus y end fraction end root f open parentheses square root of fraction numerator 4 plus y over denominator 1 minus y end fraction end root close parentheses equals fraction numerator open parentheses square root of fraction numerator 4 plus y over denominator 1 minus y end fraction end root close parentheses squared minus 4 over denominator open parentheses square root of fraction numerator 4 plus y over denominator 1 minus y end fraction end root close parentheses squared plus 1 end fraction equals fraction numerator 4 plus y minus 4 plus 4 y over denominator 4 plus y plus 1 minus y end fraction equals fraction numerator 5 y over denominator 5 end fraction equals y S o comma space f space i s space o n t o.$

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