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 then f left parenthesis x right parenthesis text  is  end text

  1. one-one and not onto

  2. one-one and onto

  3. not one-one but onto

     

  4. neither one-one nor onto

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 equals 5 x subscript 2 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 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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    l equals blank fraction numerator 3 F open parentheses fraction numerator 2 L over denominator 3 end fraction close parentheses over denominator A Y end fraction plus fraction numerator 2 F open parentheses fraction numerator L over denominator 3 end fraction close parentheses over denominator A Y end fraction
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    Graph between applied force and extension will be straight line because in elastic range
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    If the shear modulus of a wire material is 5.9blank cross times 10 to the power of 11 end exponent d y n e blank c m to the power of negative 2 end exponent then the potential energy of a wire of 4 cross times 10 to the power of 3 end exponent c m in diameter and 5 cm long twisted through an angle of 10’ , is

    To twist the wire through the angle d theta comma blankit is necessary to do the work
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    W equals blank not stretchy integral from 0 to theta of tau blank d theta equals blank not stretchy integral from 0 to theta of fraction numerator eta pi r to the power of 4 end exponent theta d theta over denominator 2 l end fraction equals blank fraction numerator eta pi r to the power of 4 end exponent theta over denominator 4 l end fraction
    W equals blank fraction numerator 5.9 blank cross times 10 to the power of 11 end exponent cross times 10 to the power of negative 5 end exponent cross times blank pi open parentheses 2 cross times 10 to the power of negative 5 end exponent close parentheses to the power of 4 end exponent pi to the power of 2 end exponent over denominator 10 to the power of negative 4 end exponent cross times 4 cross times 5 cross times 10 to the power of negative 2 end exponent cross times open parentheses 1080 close parentheses to the power of 2 end exponent end fraction
    W equals 1.253 blank cross times 10 to the power of negative 12 end exponent blank J

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    To twist the wire through the angle d theta comma blankit is necessary to do the work
    d W equals blank tau d theta
    And theta equals 10 to the power of ´ end exponent equals fraction numerator 10 over denominator 60 end fraction cross times fraction numerator pi over denominator 180 end fraction equals fraction numerator pi over denominator 1080 end fraction r a d
    W equals blank not stretchy integral from 0 to theta of tau blank d theta equals blank not stretchy integral from 0 to theta of fraction numerator eta pi r to the power of 4 end exponent theta d theta over denominator 2 l end fraction equals blank fraction numerator eta pi r to the power of 4 end exponent theta over denominator 4 l end fraction
    W equals blank fraction numerator 5.9 blank cross times 10 to the power of 11 end exponent cross times 10 to the power of negative 5 end exponent cross times blank pi open parentheses 2 cross times 10 to the power of negative 5 end exponent close parentheses to the power of 4 end exponent pi to the power of 2 end exponent over denominator 10 to the power of negative 4 end exponent cross times 4 cross times 5 cross times 10 to the power of negative 2 end exponent cross times open parentheses 1080 close parentheses to the power of 2 end exponent end fraction
    W equals 1.253 blank cross times 10 to the power of negative 12 end exponent blank J
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    The Young’s modulus of the material of a wire is equal to the

    Young’s modulus of material Y equals fraction numerator L i n e a r blank s t r e s s over denominator L o n g i t u d i n a l blank s t r a i n end fraction
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