Function Problem : 9th Feb'16

Question

Let !!A = R - {3}!! and !!B = R - {1}!!

Let !!f: A to B!! be defined by

!!f(x) = (x-2)/(x-3)!!

!!a)" "!!f is bijective

!!b)" "!!f is one-one

!!c)" "!!f is onto

!!d)" "!!one to one but not onto

Solution

To check wether !!f!! is one-one,

!!f(x) = f(y)!!

!!(x-2)/(x-3) = (y-2)/(y-3)!!

Solving this, !!x = y!!

Hence, f is one-one function

Now, !!(x-2)/(x-3) = y!!

!!=> x = (3y-2)/(y-1)!!

This shows that codomain of !!f!! belongs to range of !!f!!.

Hence, !!f!! is one-one, onto and bijective function.