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Differential Calculus

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 Joined: 6 Apr 2008 Post: 6
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

Cool goIITian

Joined: 24 Apr 2007
Posts: 88
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.

Forum Expert
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!!

Blazing goIITian

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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