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Differential Calculus

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7 Apr 2008 09:07:02 IST

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prove that:

Engineering Entrance , JEE Main , JEE Advanced , Mathematics , Differential Calculus

prove that f : RR , given by f(X)=2x, is one-one and onto

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maga ishwarya

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Joined: 24 Apr 2007

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7 Apr 2008 09:35:36 IST

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LET
            f(x₁)=f(x₂)
then
        2x₁ =2x₂
            x₁=x₂.
therefore f(x) is one-one.

LET
        f(x)=y
        y=2x
        y/2=x.
        f(y/2)=2*y/2.
               =y

therefore f(x) is onto.

Manasi

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Joined: 1 Dec 2006

Posts: 2108

7 Apr 2008 22:57:33 IST

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a function is one-one, when f(x1)=f(x2) implies x1=x2...so suppose
            f(x₁)=f(x₂)
=>        2x₁ =2x₂
=>           x₁=x₂.
thus f(x) is one-one.

again a function y=f(x )is onto, when for all possible values of 'y'...thr exists a 'x'

here y = 2x ....

i.e. x=y/2

thus here for all values of y, thr will be a 'x' defined...
therefore f(x) is onto.

nd here's a shortcut...any function is one-one and onto (both)...if it is entirely increasing (or decreasing ) in its domain....

here y=2x is a line thru the origin...always increasing, with no point of inflection...so its one-one onto fnc!!

ashish banga

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Joined: 16 Dec 2006

Posts: 1362

7 Apr 2008 23:00:02 IST

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if a line parallel to x- axis cuts the graph of the function at only one point then it is one -one function

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