IIT JEE , AIEEE Forum - Mathematics , Differential Calculus

rex_kj

prove that f : R

R , given by f(X)=2x, is one-one and onto

magaishwarya

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.

magiclko

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

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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