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Maths-

General

Easy

Question

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

not one-one
not onto
not one-one but onto
one-one but not onto

hint Hint:

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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